# Resonance between planar self-affine measures

Aleksi Pyörälä  Research Unit of Mathematical Sciences
P.O. Box 8000
FI-90014 University of Oulu
Finland<EMAIL_ADDRESS>
###### Abstract.

We show that if $\\{\phi_{i}\\}_{i\in\Gamma}$ and
$\\{\psi_{j}\\}_{j\in\Lambda}$ are self-affine iterated function systems on
the plane that satisfy strong separation, domination and irreducibility, then
for any associated self-affine measures $\mu$ and $\nu$, the inequality

$\dim_{\rm H}(\mu*\nu)<\min\\{2,\dim_{\rm H}\mu+\dim_{\rm H}\nu\\}$

implies that there is algebraic resonance between the eigenvalues of the
linear parts of $\phi_{i}$ and $\psi_{j}$. This extends to planar non-
conformal setting the existing analogous results for self-conformal measures
on the line.

###### Key words and phrases:

Self-affine measures, Hausdorff dimension, convolution of measures, resonance

###### 2020 Mathematics Subject Classification:

Primary 28A80; Secondary 37A10

I thank Ville Suomala and Meng Wu for their reading of and comments on the
manuscript.

##  1\. Introduction

In the 1960s, Furstenberg conjectured that if $X,Y\subseteq[0,1]$ are closed
sets invariant under multiplication by integers $m$ and $n$, respectively,
then for any $s\neq 0$, the inequality

(1.1) $\dim(X+sY)<\min\\{1,\dim X+\dim Y\\}$

implies that $\frac{\log m}{\log n}\in\mathbb{Q}$. Here and in the following,
$\dim$ denotes the (lower) Hausdorff dimension for both sets and measures.
This conjecture was one of several that aimed to capture the idea that if
$\frac{\log m}{\log n}\not\in\mathbb{Q}$, then expansions in base $m$ and $n$
should have no common structure: Indeed, the right-hand side of (1.1) is
always an upper bound for $\dim(X+sY)$, while the strict inequality (1.1)
implies that many of the fibers $\\{(x,y)\in X\times sY:\ x+y=z\\}$ are large,
which means that $X$ and $Y$ should have arithmetically similar structure in
many places. The phenomenon (1.1) is usually referred to as resonance: $X$ and
$Y$ are said to _resonate_ if they satisfy (1.1) for some $s$, and otherwise
they are said to _dissonate_.

It is also natural to ask if a similar phenomenon holds in a more general
setting: For dynamical systems $X$ and $Y$, does (1.1) imply some kind of
algebraic or arithmetic similarity between the sets or the dynamics? The first
result in this direction is due to Moreira [10] from 1998, who proved that for
two self-conformal sets on the line, (1.1) cannot hold in the presence of an
irrationality assumption if one of the sets is totally non-linear. Recall that
a set $K\subseteq\mathbb{R}$ is called self-conformal if

(1.2) $K=\bigcup_{i=1}^{m}f_{i}(K)$

for some $C^{1+\varepsilon}$-contractions $f_{i}$.

In 2009, Peres and Shmerkin [25] established analogous results for sums of
self-similar sets on the line: The dimension of the sum is maximal unless the
contraction ratios of the defining similarities form an _arithmetic set_. A
set $A\subseteq\mathbb{R}$ is called arithmetic if
$A\subseteq\alpha\mathbb{N}$ for some $\alpha\in\mathbb{R}$. Recall that a set
is self-similar if it satisfies (1.2) with $f_{i}$ being similarities. An
analogous result for convolutions of Cantor measures on the line was obtained
shortly after by Nazarov, Peres and Shmerkin [23]: Indeed, by replacing sum
with convolution in (1.1), one can formulate the concept of resonance for
measures.

A major breakthrough on the topic of resonance between dynamical systems was
achieved by Hochman and Shmerkin in 2012 [19], who introduced a powerful
method called the _local entropy averages_ to attack problems regarding
projections (and therefore sums) of dynamically defined fractals. Hochman and
Shmerkin managed to both prove the original conjecture of Furstenberg, and
extend to the setting of measures the existing results on the sums of self-
similar and self-conformal sets on the line.

Namely, they proved that if $\\{f_{i}\\}_{i\in\Gamma}$ and
$\\{g_{j}\\}_{j\in\Lambda}$ are families of $C^{1+\varepsilon}$-contractions
on $\mathbb{R}$ that satisfy the open set condition, then for any associated
self-conformal measures $\mu$ and $\nu$ and any $t>0$,

(1.3) $\dim(\mu*S_{t}\nu)=\min\\{1,\dim\mu+\dim\nu\\}$

unless the asymptotic contraction ratios of $\phi_{i}$ and $\psi_{j}$ form an
arithmetic set. Recall that $\mu$ is self-conformal if

(1.4) $\mu=\sum_{i\in\Gamma}p_{i}\cdot\mu\circ f_{i}^{-1}$

for a probability vector $(p_{i})_{i\in\Gamma}$ and $f_{i}$ as above. Due to
well-known variational principles, the result for self-conformal measures
implies the result for sets as well. Recently, an analogue of (1.3) was proven
by Bruce and Jin [9] for a rich class of measures on homogeneous self-similar
sets without requiring any separation conditions, in particular measures
satisfying (1.4) with $f_{i}$ being similarities. For self-conformal measures,
(1.3) was recently verified without requiring any separation conditions by
Bárány, Käenmäki, Wu and the author [8].

Perhaps surprisingly, almost nothing seems to be known of this phenomenon in
higher dimensions. While there certainly exists literature on bounding the
size of sums or convolutions from below, such as the famous inverse theorem of
Hochman [18, 17], it primarily focuses on showing that, for very general $X$
and $Y$, the sum (or convolution) $X+Y$ is _strictly larger_ than $X$, unless
$X$ and $Y$ have a very special structure. See also [27, 14] and the
references therein for progress in related phenomena. However, the existing
results do not aim to capture the spirit of the phenomenon predicted by
Furstenberg, that “geometric resonance” of dynamically defined sets should
imply a kind of “algebraic resonance” between the dynamics.

The purpose of the present work is to provide an extension of this principle
to the planar setting. However, in formulating an extension beyond the line,
one has to be careful: The direct extension, that

(1.5) $\dim(X+Y)<\min\\{2,\dim X+Y\\}$

unless there is algebraic resonance between the dynamics of $X$ and $Y$,
breaks down easily. Indeed, one can of course isometrically embed any sets $X$
and $Y$ on the line to the plane, and their sum will always have dimension at
most $1$. It is also not difficult to construct examples of $X$ and $Y$ with
dimension strictly greater than one by taking product sets. Thus, in order to
expect (1.5) to imply algebraic resonance, one has to assume that $X$ and $Y$
are “spread out” in sufficiently many directions, in some sense.

In this paper, we consider the size of $\mu*\nu$ when $\mu$ and $\nu$ are
self-affine measures on the plane, that is, they satisfy (1.4) with $f_{i}$
being invertible affine contractions on $\mathbb{R}^{2}$. Let
$\mathbb{RP}^{1}$ denote the collection of one-dimensional subspaces of
$\mathbb{R}^{2}$. For a $2\times 2$-matrix $A$, let
$|\lambda_{1}(A)|\leq|\lambda_{2}(A)|$ denote its eigenvalues. Let $A$ also
denote the action induced by $A$ on $\mathbb{RP}^{1}$. We say that a system
$\Phi=\\{f_{i}(x)=A_{i}x+a_{i}\\}_{i\in\Gamma}$ of affine contractions on
$\mathbb{R}^{2}$ satisfies

  * 1)

the _strong separation condition_ if there exists a bounded open set
$V\neq\emptyset$ such that for every $i\neq j\in\Gamma$, $f_{i}({\rm
cl}(V))\subseteq V$ and $f_{i}({\rm cl}(V))\cap f_{j}({\rm cl}(V))=\emptyset$,

  * 2)

_hyperbolicity_ if there exists at least one $i\in\Gamma$ such that
$|\lambda_{1}(A_{i})|<|\lambda_{2}(A_{i})|$,

  * 3)

_irreducibility_ if for every $\theta\in\mathbb{RP}^{1}$, there exists
$i\in\Gamma$ such that $A_{i}\theta\neq\theta$, and

  * 4)

the _domination condition_ if there exists a multicone
$\mathcal{C}\subseteq\mathbb{RP}^{1}$, i.e. a finite union of closed cones,
such that $A_{i}\mathcal{C}\subseteq{\rm int}(\mathcal{C})$ for each
$i\in\Gamma$.

######  Theorem 1.1.

Let $\Phi=\\{\varphi_{i}(x)=A_{i}x+a_{i}\\}_{i\in\Gamma}$ and
$\Psi=\\{\psi_{j}(x)=B_{j}x+b_{j}\\}_{j\in\Lambda}$ be systems of affine
contractions on $\mathbb{R}^{2}$ that satisfy the strong separation condition,
hyperbolicity, and irreducibility. Let $\mu$ and $\nu$ be fully supported
self-affine measures associated to $\Phi$ and $\Psi$, and suppose that
$\dim\mu\geq\dim\nu$. If

$\dim(\mu*\nu)<\min\\{2,\dim\mu+\dim\nu\\},$

then $\dim\mu>1>\dim\nu$ and, if $\Phi$ and $\Psi$ also satisfy the domination
condition, then

$\\{\log|\lambda_{1}(A_{i})|:\ i\in\Gamma\\}\cup\\{\log|\lambda_{2}(B_{j})|:\
j\in\Lambda\\}$

is an arithmetic set.

Let us comment on the assumptions of the theorem. The assumption of strong
separation is classical in the study of iterated function systems, since it
makes it possible to view the attractor as a dynamical system, giving access
to a multitude of tools from ergodic theory. However, during recent years,
much attention has been directed towards establishing existing results without
assuming any separation conditions, and we expect it can be removed from our
result as well.

The assumption of hyperbolicity ensures that the systems $\Phi$ and $\Psi$ are
“strictly self-affine”. This is crucial in our approach, since strictly self-
affine measures have a very special tangential structure that we will heavily
use. This structure is formulated in Proposition 4.1 which is the main
technical contribution of this paper, and we believe it can find applications
outside this work as well. Without the hyperbolicity assumption, the
contractions are similarities up to a change of basis, meaning that the self-
affine measures are essentially self-similar. Of course, it would be
interesting to find an analogous result for resonance between planar self-
similar measures.

The assumption of irreducibility is our way to ensure that the measures $\mu$
and $\nu$ are “spread out” in sufficiently many directions, which is something
one has to assume as explained in the preceding discussion. Without this
assumption, it is easy to construct examples for which the conclusion of the
theorem does not hold, by e.g. constructing measures on Bedford-McMullen
carpets. However, we do not know if it is enough to assume irreducibility for
just one of the systems $\Phi$ and $\Psi$.

If it happens that $\dim\mu\geq\dim\nu\geq 1$ or $1\geq\dim\mu\geq\dim\nu$,
then the strong separation, hyperbolicity and irreducibility are enough to
ensure that $\mu*\nu$ has the maximal dimension. The case $\dim\mu>1>\dim\nu$
is more delicate, since now $\mu$ might be large enough to “absorb” some of
$\nu$ in the convolution, and some kind of independence of their local
structures is required as in the analogous one-dimensional results. In
establishing this independence, we require the domination condition since it
gives us a way to connect the eigenvalues of $A_{i}$ and $B_{j}$ to the
dynamics of the scenery processes of $\mu$ and $\nu$ around typical points.
However, it is likely that the assumption is just a by-product of our
argument.

Finally, we remark that Theorem 1.1 combined with the variational principle of
Bárány, Käenmäki and Rossi [5, Proposition 2.4] yields an analogous result for
sets:

######  Corollary 1.2.

Let $\Phi=\\{\varphi_{i}(x)=A_{i}x+a_{i}\\}_{i\in\Gamma}$ and
$\Psi=\\{\psi_{j}(x)=B_{j}x+b_{j}\\}_{j\in\Lambda}$ be systems of affine
contractions on $\mathbb{R}^{2}$ that satisfy the strong separation condition,
domination, and irreducibility. Let $X$ and $Y$ denote the attractors of
$\Phi$ and $\Psi$, and suppose that $\dim X\geq\dim Y$. If

$\dim(X+Y)<\min\\{2,\dim X+\dim Y\\},$

then $\dim X>1>\dim Y$ and there exists $n\in\mathbb{N}$ such that

$\\{\log|\lambda_{1}(A_{i_{1}}\ldots A_{i_{n}})|:\
(i_{1},\ldots,i_{n})\in\Gamma^{n}\\}\cup\\{\log|\lambda_{2}(B_{j_{1}}\ldots
B_{j_{n}})|:\ (j_{1},\ldots,j_{n})\in\Lambda^{n}\\}$

is an arithmetic set.

###### Proof.

Suppose that $\dim(X+Y)<\min\\{2,\dim X+\dim Y\\}$ and let $\varepsilon>0$ be
small. Using [5, Proposition 2.4], choose $n\in\mathbb{N}$ and fully supported
self-affine measures $\mu$ and $\nu$ associated to the systems
$\\{\varphi_{i_{1}}\circ\cdots\circ\varphi_{i_{n}}\\}_{(i_{1},\ldots,i_{n})\in\Gamma^{n}}$
and
$\\{\psi_{j_{1}}\circ\cdots\circ\psi_{j_{n}}\\}_{(j_{1},\ldots,j_{n})\in\Gamma^{n}}$,
such that $\dim\mu\geq\dim X-\varepsilon$ and $\dim\nu\geq\dim Y-\varepsilon$.
Then, since $\mu*\nu$ is supported on $X+Y$, for small enough $\varepsilon$ we
have

$\dim(\mu*\nu)\leq\dim(X+Y)<\min\\{2,\dim X+\dim
Y\\}-3\varepsilon\leq\min\\{2,\dim\mu+\dim\nu\\}-\varepsilon.$

The conclusion regarding eigenvalues now follows from Theorem 1.1. In
addition, we have $\dim\mu>1>\dim\nu$, which implies that $\dim X>1\geq\dim Y$
if $\varepsilon$ was chosen small enough. If it were that $\dim Y=1$, then
$\dim\mu>1\geq\dim\nu\geq 1-\varepsilon$, and Claim 4.2 combined with Theorem
3.2 would assert that $\dim(\mu*\nu)>2-\varepsilon$ which is a contradiction.
Thus $\dim Y<1$ and the proof is complete. ∎

###  1.1. On the proof of Theorem 1.1

For simplicity, we suppose that the contractions in $\Phi$ and $\Psi$ map the
unit ball into disjoint ellipses. While stronger than the separation condition
that we assume, the same argument works for the classical strong separation
condition up to minor technical additions.

We will prove the statement by contradiction: We show that if any of the
conditions

  1. i)

$\dim\mu\geq\dim\nu\geq 1$,

  2. ii)

$1\geq\dim\mu\geq\dim\nu$, or

  3. iii)

$\\{\log|\lambda_{1}(A_{i})|:\ i\in\Gamma\\}\cup\\{\log|\lambda_{2}(B_{j})|:\
j\in\Lambda\\}$ is not an arithmetic set

holds, then we will have $\dim(\mu*\nu)=\min\\{2,\dim\mu+\dim\nu\\}$. Our
proof is based on the local entropy averages of Hochman-Shmerkin [19]: instead
of proving directly that $\dim(\mu*\nu)$ is large, we will show that the
entropy of $\mu^{\prime}*\nu^{\prime}$ is large on many scales, where
$\mu^{\prime}$ and $\nu^{\prime}$ are _magnifications_ of $\mu$ and $\nu$
along properly chosen filtrations of their supports.

The machinery of Hochman-Shmerkin [19] asserts that $\dim(\mu*\nu)$ is at
least the average of finite-scale entropies of magnifications of $\mu$ and
$\nu$. With this machinery, the cases i) and ii) above are very simple in
principle. Indeed, magnifying both $\mu$ and $\nu$ along the “cylinder
ellipses” $\varphi_{i_{1}}\circ\cdots\circ\varphi_{i_{n}}(B(0,1))$ and
$\psi_{j_{1}}\circ\cdots\circ\psi_{j_{n}}(B(0,1))$, on the limit they both
resemble orthogonal projections of the original measures, by the hyperbolicity
assumption. As such, they will have entropy close to either $1$ in the case
i), or $\dim\mu$ and $\dim\nu$ in the case ii), by the strong projection
theorem for self-affine measures [2, Theorem 7.1], due to Bárány, Hochman and
Rapaport. Since the ellipses on which the magnifications are supported on have
major axes pointing in different directions by the irreducibility assumption,
their convolution has a product-like structure on the plane and thus has
entropy close to $\min\\{2,\dim\mu+\dim\nu\\}$.

The situation $\dim\mu>1>\dim\nu$ is more delicate, since the magnifications
of $\mu$ along the cylinder ellipses are no longer enough to store a
sufficient amount of the dimension of $\mu$. Instead, we will apply the local
entropy averages with $\mu$ magnified along dyadic squares and $\nu$ along the
cylinder ellipses.

For any such magnifications $\mu^{\prime}$ and $\nu^{\prime}$ of $\mu$ and
$\nu$, and any orthogonal projection $\pi$, applying the chain rule of entropy
yields that the entropy of $\mu^{\prime}*\nu^{\prime}$ is equal to

$\displaystyle\text{entropy of}\ \pi\mu^{\prime}*\pi\nu^{\prime}\ +\
\text{conditional entropy of}\ \mu^{\prime}*\nu^{\prime}\ \text{w.r.t.}\ \pi$
(1.6) $\displaystyle\geq\ $ $\displaystyle\text{entropy of}\
\pi\mu^{\prime}*\pi\nu^{\prime}\ +\ \text{entropy of}\
\mu^{\prime}-\text{entropy of}\ \pi\mu^{\prime}.$

The key geometric ingredient in our proof is the observation that each
$\mu^{\prime}$ has a _fiber structure_ in the sense that $\pi\mu^{\prime}$ is
(close to) a _slice measure_ of $\mu$, for a properly chosen $\pi$. The
precise form of this structure is stated in Proposition 4.1. Similar fiber
structures have been previously observed for self-affine sets by Käenmäki,
Koivusalo and Rossi [21], for self-affine measures on Bedford-McMullen carpets
by Ferguson, Fraser and Sahlsten [13] and for self-affine measures with an
additional projection condition by Kempton [22]. Combining this with the
dimension conservation phenomenon that follows for planar self-affine measures
from the Ledrappier-Young formula of Bárány [1] and the general fact that the
average entropy of $\mu^{\prime}$ over many scales is close to the dimension
of $\mu$, we see that upon averaging, (1.1) is in fact close to

(1.7) $\text{entropy of}\ \pi\mu^{\prime}*\pi\nu^{\prime}\ +\ 1,$

recalling that we are assuming $\dim\mu>1$.

Thus, if we manage to show that for the measures $\pi\mu^{\prime}$ and
$\pi\nu^{\prime}$ _on the line_ we have

(1.8) $\text{entropy of}\
\pi\mu^{\prime}*\pi\nu^{\prime}\geq\min\\{1,\text{entropy of}\
\pi\mu^{\prime}\ +\ \text{entropy of}\ \pi\nu^{\prime}\\}-o(1),$

then by (1.7), we have that the average entropy of $\mu^{\prime}*\nu^{\prime}$
is at least

$\displaystyle\min\\{2,\text{entropy of}\ \pi\mu^{\prime}\ +\ 1\ +\
\text{entropy of}\ \pi\nu^{\prime}\\}-o(1)$ $\displaystyle=\ $
$\displaystyle\min\\{2,\dim\mu+\dim\nu\\}-o(1)$

completing the proof by the local entropy averages and another application of
the dimension conservation of [1] and the projection theorem of [2].

Proving (1.8) is the part where the assumption iii) on the eigenvalues of
$A_{i}$ and $B_{j}$ steps in. We need to inspect the dynamics of the sequences
$(\pi\mu^{\prime})$ and $(\pi\nu^{\prime})$ obtained by continuously
magnifying $\mu$ and $\nu$, and show that they are independent of each other
in a sense. Since $\pi\mu^{\prime}$ is (close to) a slice of $\mu$ and
$\pi\nu^{\prime}$ is an orthogonal projection of $\nu$, they both have
“dynamically self-similar” structure, which allows us to study the
independence of $(\pi\mu^{\prime})$ and $(\pi\nu^{\prime})$ via mixing
properties of certain underlying dynamical systems. Although this part bears
some resemblance to the works on convolutions of self-similar measures on the
line, new ideas are required due to the underlying dynamics being more
complicated.

###  1.2. Structure

In Section 2 we introduce our setting more rigorously, and collect some
general known results and short lemmas on self-affine measures, dynamical
systems and linear algebra. Section 3 is devoted to translating the local
entropy averages machinery of [19] to our setting, while in Section 4 we state
our main geometric and dynamical results, and explain how the proof of Theorem
1.1 is concluded using these. Section 5 is perhaps the most technical one,
devoted to the proof of the main geometric result, Proposition 4.1. The
arguments here were inspired by the work of Kempton [22]. Finally, in Section
6 we investigate the dynamics of the sequences of magnifications of $\mu$ and
$\nu$, and prove the required lower bounds for their average entropies.

Table 1. Notation $\Gamma,\Lambda$ |  Finite alphabets
---|---
$\mathtt{i},\mathtt{j},\mathtt{k},\ldots$ |  Infinite words of $\Gamma^{\mathbb{N}}$, $\Lambda^{\mathbb{N}}$
$\mathtt{a},\mathtt{b},\ldots$ |  Finite words
$\Phi=\\{\varphi_{i}\\}_{i\in\Gamma},\Psi=\\{\psi_{j}\\}_{j\in\Lambda}$ |  Systems of affine invertible contractions
$\bar{\mu},\bar{\nu}$ |  Bernoulli measures on $\Gamma^{\mathbb{N}}$ and $\Lambda^{\mathbb{N}}$
$\Pi$ |  The natural projections $\Gamma^{\mathbb{N}}\rightarrow\mathbb{R}^{2}$ and $\Lambda^{\mathbb{N}}\rightarrow\mathbb{R}^{2}$
$\mu,\nu$ |  Projections of $\bar{\mu}$, $\bar{\nu}$ through $\Pi$
$\mu_{D}$ |  Normalized restricition on $D$
$\mu^{D}$ |  Measure $\mu_{D}$ linearly rescaled onto $[-1,1)^{d}$
$\pi_{\theta}$ |  Orthogonal projection onto the line $\theta$
$\pi^{i}$ |  Projection onto the $i$th coordinate
$\mu_{\mathtt{i},\theta}$ |  A “slice measure” of $\mu$; see (4.1)
$S_{t},T_{x}$ |  Scaling by $2^{t}$ and translation by $-x$, respectively
$R_{\theta}$ |  A rotation taking $\theta$ onto the $y$-axis
$Y_{\mathtt{i},\theta,r_{1},r_{2}}$ |  A rectangle with side lengths $2^{-r_{2}}\leq 2^{-r_{1}}$
$H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}$ |  An affine map rescaling $Y_{\mathtt{i},\theta,r_{1},r_{2}}$ onto $R_{\theta}^{-1}[-1,1]^{2}$
$A=UDV^{-1}$ |  The singular value decomposition
$L_{\mathtt{i},\theta,k},F_{\theta,\mathtt{i}}^{k}$ |  Non-singular linear maps $\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$
$Q_{\mathtt{i},\theta,k}$ |  $Y_{\sigma^{\ell_{k}}\mathtt{i},\theta,kN+\log\alpha_{1}(\mathtt{i}|_{\ell_{k}}),kN+\log\alpha_{2}(\mathtt{i}|_{\ell_{k}})}$
$E_{\mathtt{i},\theta,k}$ |  The largest ellipse contained in $Q_{\mathtt{i},\theta,k}$
$\theta(A)$ |  The direction of the longer axis of $A(B(0,1))$
$\theta(\mathtt{i}),\theta^{-}(\mathtt{i}),\theta^{*}(\mathtt{i})$ |  Limit orientations; see Lemma 2.1
$\|A\|$ |  Operator norm of $A$
$A|_{\theta}$ |  The restriction of $A$ onto the line $\theta$
$\alpha_{1}(A)\leq\alpha_{2}(A)$ |  The singular values of $A$
$|\lambda_{1}(A)|\leq|\lambda_{2}(A)|$ |  The eigenvalues of $A$
$i_{k}(\mathtt{i}),\ i_{k}(\mathtt{j})$ |  Stopping times; $\|A_{\mathtt{i}|_{i_{k}}}\|\approx\|B_{\mathtt{i}|_{i_{k}}}\|\approx 2^{-kN}$
$\ell_{k}=\ell_{k}(\mathtt{i})$ |  An increasing sequence; see (5.1)
$\mu_{\mathtt{i}|_{i_{k}}},\ \nu_{\mathtt{j}|_{i_{k}}},\ \mu^{\mathtt{i},k}$ |  Magnifications of $\mu$ and $\nu$; see Notation 3.1
$\rho(n,(\mathtt{i},\theta))$ |  The reflection done by $A_{\mathtt{i}|_{n}}^{-1}$ on the line $\theta$
$\mathcal{Z}_{\Phi}=(Z_{\Phi},\lambda_{\Phi},\mathcal{T}_{s})$ |  A flow related to $(\pi_{2}\mu^{\mathtt{i},k})_{k\in\mathbb{N}}$; see Section 6
$\mathcal{Z}_{\Psi}=(Z_{\Psi},\lambda_{\Psi},\mathcal{T}_{s})$ |  A flow related to $(\pi_{2}\nu_{\mathtt{j}|_{i_{k}}})_{k\in\mathbb{N}}$; see Section 6
$\mathcal{Z}_{\Phi}^{\prime},\mathcal{Z}_{\Psi}^{\prime}=(Z_{\Psi}^{\prime},\lambda_{\Psi}^{\prime},\mathcal{T}_{s})$ |  Projections of $\mathcal{Z}_{\Phi}$ and $\mathcal{Z}_{\Psi}$ through $\pi^{1,2,4}$
$F:\mathcal{Z}_{\Phi}\rightarrow\mathcal{P}(\mathbb{R}^{2})$ |  Coding of $\pi^{2}\mu^{\mathtt{i},k}$ via $\mathcal{Z}_{\Phi}$; see (6.5)
$G:\mathcal{Z}_{\Psi}\rightarrow\mathcal{P}(\mathbb{R}^{2})$ |  Coding of $\pi^{2}\nu_{\mathtt{j}|_{i_{k}}}$ via $\mathcal{Z}_{\Psi}$; see (6.7)
$F^{\prime},G^{\prime}$ |  Functions $F$ and $G$ without reflection

##  2\. Preliminaries

In this paper, a measure refers to a Radon measure on a metrizable topological
space. The notation $\mathcal{P}(X)$ stands for probability measures on the
space $X$. For a measure $\mu$ on $X$ and a subset $Y\subseteq X$, $\mu|_{Y}$
denotes the restriction of $\mu$ onto $Y$, $\mu_{Y}:=\mu(Y)^{-1}\mu|_{Y}$ the
normalized restriction when $\mu(Y)>0$. For a measurable function $f$, let
$f\mu:=\mu\circ f^{-1}$ denote the push-forward. The space of probability
measures is always equipped with the weak-∗ topology which we metrize using
the Lévy-Prokhorov metric $d_{\rm LP}$,

$d_{\rm LP}(\mu,\nu)=\inf\\{\varepsilon>0:\
\mu(A)\leq\nu(A^{\varepsilon})+\varepsilon,\
\nu(A)\leq\mu(A^{\varepsilon})+\varepsilon\ \text{for all Borel}\ A\\},$

where $A^{\varepsilon}$ denotes the open $\varepsilon$-neighbourhood of $A$.
We measure the size of measures with the lower Hausdorff dimension,

$\dim\mu=\operatorname{\dim_{H}}\mu=\inf\\{\operatorname{\dim_{H}}(E):\
\mu(E)>0\\},$

and occasionally with upper and lower local dimensions,

$\underline{\dim}_{\rm loc}\mu(x)=\liminf_{r\rightarrow
0}\frac{\log\mu(B(x,r))}{\log r},\qquad\overline{\dim}_{\rm
loc}\mu(x)=\limsup_{r\rightarrow 0}\frac{\log\mu(B(x,r))}{\log r},$

where $B(x,r)$ denotes the closed ball centered at $x$ and of radius $r$. The
measure $\mu$ is called exact dimensional if $\underline{\dim}_{\rm
loc}\mu(x)=\overline{\dim}_{\rm loc}\mu(x)$ almost everywhere. The following
connection between Hausdorff and lower local dimension is well-known: For any
measure $\mu$,

$\dim\mu={{\rm ess}\inf}_{x\sim\mu}\underline{\dim}_{\rm loc}\mu(x).$

###  2.1. Symbolic dynamics

Let $\Gamma$ be a finite set with $\\#\Gamma\geq 2$, and let
$\Phi=\\{\varphi_{i}\\}_{i\in\Gamma}$ be an iterated function system of
contractions on $\mathbb{R}^{d}$. It is well-known that there exists a unique
compact set $K$, called the _attractor_ of $\Phi$, such that

$K=\bigcup_{i\in\Gamma}\varphi_{i}(K).$

We refer to Falconer’s book [11] for standard properties of iterated function
systems. If the functions $\varphi_{i}$ are of the form
$\varphi_{i}(x)=A_{i}x+a_{i}$, where
$A_{i}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ are invertible linear maps
with $\|A_{i}\|<1$ and $a_{i}\in\mathbb{R}^{d}$, then the IFS $\Phi$ is called
_self-affine_ and its attractor a self-affine set.

Write $\Gamma^{*}=\bigcup_{n}\Gamma^{n}$ for the set of finite words composed
of elements of $\Gamma$. For a finite word $\mathtt{a}=i_{0}i_{1}\ldots
i_{n}\in\Gamma^{*}$ we write
$\varphi_{\mathtt{a}}=\varphi_{i_{0}}\circ\cdots\circ\varphi_{i_{n}}$. Let
$|\mathtt{a}|$ denote the number of elements in $\mathtt{a}$. For finite words
$\mathtt{a}$ and $\mathtt{b}$, let
$\mathtt{a}\mathtt{b}\in\Gamma^{|\mathtt{a}|+|\mathtt{b}|}$ denote their
concatenation.

For a word $\mathtt{i}\in\Gamma^{\mathbb{N}}\cup\Gamma^{*}$ and and integer
$k\leq|\mathtt{i}|$, let $\mathtt{i}|_{k}\in\Gamma^{k}$ denote its projection
to the first $k$ coordinates. When $k=0$, set $\mathtt{i}|_{k}=\emptyset$. For
$\mathtt{i},\mathtt{j}\in\Gamma^{\mathbb{N}}$, let
$\mathtt{i}\wedge\mathtt{j}:=\mathtt{i}|_{k}$, where $k$ is the largest
integer for which $\mathtt{i}|_{k}=\mathtt{j}|_{k}$. Define a distance $d$ on
$\Gamma^{\mathbb{N}}$ by

$d(\mathtt{i},\mathtt{j})=2^{-|\mathtt{i}\wedge\mathtt{j}|}$

for every $\mathtt{i},\mathtt{j}\in\Gamma^{\mathbb{N}}$. For a finite word
$\mathtt{a}\in\Gamma^{*}$, write $[\mathtt{a}]$ for the cylinder set
$\\{\mathtt{i}\in\Gamma^{\mathbb{N}}:\
\mathtt{i}|_{|\mathtt{a}|}=\mathtt{a}\\}$. It is not difficult to see that the
cylinder sets are closed and open in the topology generated by $d$.

It is sometimes convenient to consider the two-sided sequence space
$\Gamma^{\mathbb{Z}}$. For $\mathtt{i}=\ldots
i_{-2}i_{-1};i_{0}i_{1}i_{2}\ldots\in\Gamma^{\mathbb{Z}}$ and $m\leq
n\in\mathbb{Z}$, write $\mathtt{i}|_{m}^{n}=i_{m}i_{m+1}\ldots i_{n}$. The
metric $d$ extends to $\Gamma^{\mathbb{Z}}$ by replacing $\mathtt{i}|_{k}$ by
$\mathtt{i}|_{-k}^{k}$ in the definition of $\mathtt{i}\wedge\mathtt{j}$. The
cylinder sets of $\Gamma^{\mathbb{Z}}$ are given by
$[\mathtt{i}]_{m}^{n}:=\\{\mathtt{j}\in\Gamma^{\mathbb{Z}}:\
\mathtt{j}|_{m}^{n}=\mathtt{i}|_{m}^{n}\\}$. There is a natural surjection
$\Gamma^{\mathbb{Z}}\rightarrow\Gamma^{\mathbb{N}}$ given by the restriction
to the “positive coordinates”, $\ldots
i_{-1};i_{0}i_{1}\ldots=:\mathtt{i}\mapsto\mathtt{i}^{+}:=i_{0}i_{1}\ldots$.
Similarly, we define the projection to the “negative coordinates” by
$\mathtt{i}^{-}:=i_{-1}i_{-2}\ldots\in\Gamma^{\mathbb{N}}$.

We let $\sigma$ denote the left-shift on both $\Gamma^{\mathbb{N}}$ and
$\Gamma^{\mathbb{Z}}$, given by $\sigma(i_{0}i_{1}\ldots)=i_{1}i_{2}\ldots$
and $\sigma(i_{-1};i_{0}i_{1}\ldots)=\ldots i_{0};i_{1}i_{2}\ldots$. The
tuples $(\Gamma^{\mathbb{N}},\sigma)$ and $(\Gamma^{\mathbb{Z}},\sigma)$ are
referred to as the one-sided and two-sided shift spaces, respectively. For any
$\sigma$-invariant probability measure $\nu$ on $\Gamma^{\mathbb{Z}}$, there
is a unique $\sigma$-invariant probability measure on $\Gamma^{\mathbb{Z}}$
which we also denote by $\nu$, given by
$\nu([\mathtt{i}]_{m}^{n}):=\nu([\mathtt{i}|_{m}^{n}])$ for each
$\mathtt{i}\in\Gamma^{\mathbb{Z}}$, $m\leq n\in\mathbb{Z}$. This is referred
to as the natural extension of $\nu$.

###  2.2. Linear algebra and matrix products

Let $A$ be a real-valued $2\times 2$-matrix. Recall that the eigenvalues of
$A$ are denoted by $\lambda_{1}(A)$ and $\lambda_{2}(A)$ with
$|\lambda_{1}(A)|\leq|\lambda_{2}(A)|$. The matrix $A$ is called _hyperbolic_
(or _proximal_) if it has two real eigenvalues of different absolute value. If
a collection of matrices satisfies the domination condition, then each of the
matrices is hyperbolic; see [4, Corollary 2.4]. A hyperbolic matrix $A$ maps
the unit ball onto an ellipse with major axes of length
$\alpha_{1}(A)<\alpha_{2}(A)$, where $\alpha_{1}(A),\alpha_{2}(A)$ are the
singular values of $A$. Let $\theta(A)\in\mathbb{RP}^{1}$ denote the line
parallel to the longer semiaxis of this ellipse.

For a finite word $\mathtt{a}=i_{0}i_{1}\ldots i_{n}\in\Gamma^{*}$, write
$A_{\mathtt{a}}=A_{i_{0}}A_{i_{1}}\ldots A_{i_{n}}$ and
$A_{\mathtt{a}}^{-1}=(A_{\mathtt{a}})^{-1}=A_{i_{n}}^{-1}\ldots
A_{i_{1}}^{-1}A_{i_{0}}^{-1}$. We define a distance $d$ on $\mathbb{RP}^{1}$,
given by the smaller angle between lines. The following lemma is, for us, the
key technical consequence of the domination assumption in Theorem 1.1.

######  Lemma 2.1.

Let $\\{A_{i}\\}_{i\in\Gamma}$ be a collection of $2\times 2$-matrices
satisfying the domination condition. Then the limits

$\displaystyle\theta(\mathtt{i})$
$\displaystyle:=\lim_{n\rightarrow\infty}\theta(A_{\mathtt{i}|_{n}}),$
$\displaystyle\theta^{-}(\mathtt{i})$
$\displaystyle:=\lim_{n\rightarrow\infty}\theta(A_{i_{0}}^{-1}A_{i_{1}}^{-1}\ldots
A_{i_{n}}^{-1}),$ $\displaystyle\theta^{*}(\mathtt{i})$
$\displaystyle:=\lim_{n\rightarrow\infty}\theta(A_{i_{0}}^{*}A_{i_{1}}^{*}\ldots
A_{i_{n}}^{*})$

exist for every $\mathtt{i}\in\Gamma^{\mathbb{N}}$, the convergences are
uniform, and the functions $\mathtt{i}\mapsto\theta(\mathtt{i})$,
$\mathtt{i}\mapsto\theta^{-}(\mathtt{i})$ and
$\mathtt{i}\mapsto\theta^{*}(\mathtt{i})$ are Hölder continuous.

###### Proof.

See [26, Lemma 2.1] and [21, Lemma 2.1]. ∎

######  Lemma 2.2.

For any $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and
$\theta\in\mathbb{RP}^{1}\setminus\\{\theta(\mathtt{i})\\}$,

$\lim_{n\rightarrow\infty}d(A_{\mathtt{i}|_{n}}^{-1}\theta,\
\theta(A_{\mathtt{i}|_{n}}^{-1}))\rightarrow 0.$

In particular, $\theta(\Gamma^{\mathbb{N}})\subseteq\mathcal{C}$, where
$\mathcal{C}$ is the strongly invariant multicone of
$\\{A_{i}\\}_{i\in\Gamma}$.

###### Proof.

Let
$A_{\mathtt{i}|_{n}}=U_{\mathtt{i}|_{n}}D_{\mathtt{i}|_{n}}V_{\mathtt{i}|_{n}}^{-1}$
denote the singular value decomposition. By Lemma 2.1, $U_{\mathtt{i}|_{n}}$
takes the $x$-axis onto a line which tends to $\theta(\mathtt{i})$ as
$n\rightarrow\infty$. In particular, $U_{\mathtt{i}|_{n}}^{-1}\theta$ remains
uniformly bounded away from the $x$-axis. By [6, Theorem B],
$\frac{\alpha_{1}(A_{\mathtt{i}|_{n}})}{\alpha_{2}(A_{\mathtt{i}|_{n}})}\rightarrow
0$, whence $U_{\mathtt{i}|_{n}}^{-1}\theta$ is pulled very close to the
$y$-axis by $D_{\mathtt{i}|_{n}}^{-1}$ and finally taken close to
$\theta(A_{\mathtt{i}|_{n}}^{-1})$ by $V_{\mathtt{i}|_{n}}^{-1}$.

For the second statement, note that
$A_{\mathtt{i}|_{n}}\mathcal{C}\subseteq\mathcal{C}$ and
$d(A_{\mathtt{i}|_{n}}\mathcal{C},\theta(A_{\mathtt{i}|_{n}}))\rightarrow 0$
as $n\rightarrow\infty$ by the above, since $\mathcal{C}$ contains more than
one point. Since $\mathcal{C}$ is closed, the claim follows. ∎

We require the following technical observation, that the directions
$\theta(\mathtt{i})^{\perp}$ and $\theta^{*}(\mathtt{i})$ are bounded away
from each other, uniformly in $\mathtt{i}$.

######  Lemma 2.3.

Let $\\{A_{i}\\}_{i\in\Gamma}$ be a dominated tuple of matrices. Then there
exist a constant $C>0$ such that

$d(\theta(\mathtt{i})^{\perp},\ \theta^{*}(\mathtt{i}))\geq C$

for every $\mathtt{i}\in\Gamma^{\mathbb{N}}$.

###### Proof.

Let $f$ denote the function $\mathtt{i}\mapsto d(\theta(\mathtt{i})^{\perp},\
\theta^{*}(\mathtt{i}))$. Since
$d(\theta(\mathtt{i})^{\perp},\theta^{*}(\mathtt{i}))=d(\theta(\mathtt{i}),\theta^{*}(\mathtt{i})^{\perp})$,
and it is not difficult to see from the singular value decomposition that
$\theta^{*}(\mathtt{i})=\theta^{-}(\mathtt{i})^{\perp}$, we have that
$f(\mathtt{i})=d(\theta(\mathtt{i}),\ \theta^{-}(\mathtt{i}))$. Since $f$ is
continuous by Lemma 2.1 and $\Gamma^{\mathbb{N}}$ is compact, it suffices to
show that $f(\mathtt{i})\neq 0$ for every $\mathtt{i}\in\Gamma^{\mathbb{N}}$.

By the assumption, $\\{A_{i}\\}_{i\in\Gamma}$ has a strongly invariant
multicone $\mathcal{C}$. By Lemma 2.2, $\theta(\mathtt{i})\in{\rm
int}(\mathcal{C})$ for every $\mathtt{i}\in\Gamma^{\mathbb{N}}$. On the other
hand,
$A_{i}^{-1}(\mathbb{RP}^{1}\setminus\mathcal{C})\subseteq\mathbb{RP}^{1}\setminus\mathcal{C}$
for every $i\in\Gamma$, whence $\theta^{-}(\mathtt{i})\in{\rm
cl}(\mathbb{RP}^{1}\setminus\mathcal{C})$ for every
$\mathtt{i}\in\Gamma^{\mathbb{N}}$.

In particular, $\theta(\mathtt{i})\neq\theta^{-}(\mathtt{i})$ and the result
follows. ∎

For a matrix $A$, let $v_{i}(A)$ denote the eigenspace associated to
$\lambda_{i}(A)$ for $i=1,2$. When $A$ is hyperbolic,
$v_{i}(A)\in\mathbb{RP}^{1}$ for $i=1,2$. The following observation follows
immediately from the eigenvalue decomposition.

######  Lemma 2.4.

For a hyperbolic matrix $A$,
$\lim_{n\rightarrow\infty}\theta(A^{n})=v_{2}(A)$.

###  2.3. The Furstenberg measures

For any probability vector $(p_{i})_{i\in\Gamma}$ and Bernoulli measure
$\bar{\mu}:=p^{\mathbb{N}}$ on $\Gamma^{\mathbb{N}}$, there are associated
measures $\mu_{F}$ and $\mu_{F}^{*}$ on $\mathbb{RP}^{1}$, called the
_Furstenberg measures_ , which are given by

$\displaystyle\mu_{F}$
$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\delta_{\theta(A_{i_{n}}^{-1}\ldots
A_{i_{0}}^{-1})},$ $\displaystyle\mu_{F}^{*}$
$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\delta_{\theta(A_{i_{n}}^{*}\ldots
A_{i_{0}}^{*})}$

for $\bar{\mu}$-almost every $\mathtt{i}$. The measures $\mu_{F}$ and
$\mu_{F}^{*}$ are supported on the sets $\theta^{-}(\Gamma^{\mathbb{N}})$ and
$\theta^{*}(\Gamma^{\mathbb{N}})=\theta(\Gamma^{\mathbb{N}})^{\perp}$,
respectively. The product measures $\bar{\mu}\times\mu_{F}$ and
$\bar{\mu}\times\mu_{F}^{*}$ are invariant and ergodic under the maps

$\displaystyle
M:(\mathtt{i},\theta)\mapsto(\sigma\mathtt{i},A_{i_{0}}^{-1}\theta),$
$\displaystyle
M_{*}:(\mathtt{i},\theta)\mapsto(\sigma\mathtt{i},A_{i_{0}}^{*}\theta),$

respectively. When $\Phi$ satisfies the domination condition, the measures
$\bar{\mu}\times\mu_{F}$ and $\bar{\mu}\times\mu_{F}^{*}$ are images of the
Bernoulli measure $p^{\mathbb{Z}}$ on $\Gamma^{\mathbb{Z}}$ through the factor
maps $\mathtt{i}\mapsto(\mathtt{i}^{+},\theta^{-}(\mathtt{i}^{-}))$ and
$\mathtt{i}\mapsto(\mathtt{i}^{+},\theta^{*}(\mathtt{i}^{-}))$, which easily
implies the existence, invariance and ergodicity of $\bar{\mu}\times\mu_{F}$
and $\bar{\mu}\times\mu_{F}^{*}$. These measures do also exist without the
domination assumption, with the prescribed properties, which is a classical
result of Furstenberg. When $\Phi$ is irreducible, it is known that the
measures $\mu_{F}$ and $\mu_{F}^{*}$ are non-atomic.

###  2.4. Self-affine measures

Recall that $K$ denotes the attractor of the system of affine contractions
$\Phi$. Let $\Pi:\Gamma^{\mathbb{N}}\rightarrow K$ denote the surjection

$\mathtt{i}\mapsto\lim_{k\rightarrow\infty}\varphi_{\mathtt{i}|_{k}}(0)$

which we call the natural projection. Fix a probability vector
$p=(p_{i})_{i\in\Gamma}$ and let $\bar{\mu}=p^{\mathbb{N}}$ denote the
Bernoulli measure on $\Gamma^{\mathbb{N}}$ with marginal $p$. The measure
$\mu:=\Pi\bar{\mu}=\bar{\mu}\circ\Pi^{-1}$ is called the _self-affine measure_
associated to $p$, and is well-known to be a Radon measure supported on $K$
that satisfies

$\mu=\sum_{i\in\Gamma}p_{i}\cdot\varphi_{i}\mu.$

Throughout the paper, we will use $\dim$ to denote the Hausdorff dimension of
both sets and measures. The following strong projection theorem for self-
affine measures is due to by Bárány, Hochman and Rapaport [2]. For
$\theta\in\mathbb{RP}^{1}$, let $\pi_{\theta}:\mathbb{R}^{2}\rightarrow\theta$
denote the orthogonal projection onto the line $\theta$.

######  Theorem 2.5 ([2], Theorem 7.1).

Let $\mu$ be a self-affine measure associated to a totally irreducible,
hyperbolic IFS $\Phi$ with the strong separation condition. Then for any
orthogonal projection $\pi$,

$\dim\pi\mu=\min\\{1,\dim\mu\\}.$

We remark that the result was proved earlier for non-hyperbolic IFSs by
Hochman and Shmerkin in [19].

It follows from the Ledrappier-Young formula due to Bárány [3] that planar
self-affine measures satisfy dimension conservation in directions typical for
the Furstenberg measure $\mu_{F}$. The Ledrappier-Young formula was shortly
after generalized to higher dimensions by Bárány and Käenmäki [3]. The
application to dimension conservation was recently generalized by Feng [12] to
higher dimensions and for more general self-affine measures. Let
$\mu=\int\mu_{x}^{\theta}\,d\pi_{\theta}\mu(x)$ denote the disintegration of
$\mu$ with respect to the orthogonal projection $\pi_{\theta}$.

######  Theorem 2.6 (Corollary of [1], Theorem 2.7).

Let $\mu$ be a self-affine measure associated to an affine system of
contractions $\Phi$ with the strong separation and domination conditions, and
$\mu_{F}$ the associated Furstenberg measure. Then for $\mu_{F}$-a.e. $\theta$
and $\pi_{\theta}\mu$-a.e. $x$, the measure $\mu_{x}^{\theta}$ is exact
dimensional and

$\dim\mu=\dim\pi_{\theta^{\perp}}\mu+\dim\mu^{\theta}_{x}.$

###  2.5. Flows and eigenvalues

Let $I$ be either $\mathbb{R}$ or $[0,+\infty)$, and let $\\{T_{s}\\}_{s\in
I}$ be a family of measurable functions on a measure space $(X,\mu)$ with the
property that $T_{s}\circ T_{t}=T_{s+t}$ for each $s,t\in I$. Recall that a
real number $c\in\mathbb{R}$ is an _eigenvalue_ of the flow $(X,T_{s},\mu)$ if
there exists a measurable function $f:X\rightarrow\mathbb{C}$ such that
$f\circ T_{s}=e(cs)f$ almost everywhere, for every $s\in I$, where we write
$e(x):=\exp(2\pi ix)$. If the flow is measure-preserving, i.e. $T_{s}\mu=\mu$
for every $s$, then it is not difficult to see that any eigenfunction $f$
satisfies $|f|\equiv 1$. We now record some standard properties of
eigenvalues.

######  Lemma 2.7 (Special case of Lemma 3.11, [16]).

Let $(X,T_{s},\mu)$ be an ergodic measure-preserving flow, and let
$c\in\mathbb{R}$. Then the discrete-time dynamical system $(X,T_{c},\mu)$ is
ergodic if and only if $c^{-1}$ is not an eigenvalue of $(X,T_{s},\mu)$.

######  Lemma 2.8.

A measure-preserving flow $(X,T_{s},\mu)$ on a standard Borel space $X$ has at
most countably many eigenvalues.

###### Proof.

It is not difficult to see that eigenfunctions for different eigenvalues are
orthogonal. Since the space $L^{2}(X)$ is separable, any collection of
orthogonal functions has to be countable. ∎

The proof of the following result for the product of discrete-time dynamical
systems can be found in many textbooks. Most proofs utilize the spectral
theorem, and applying the spectral theorem for one-parameter families of
unitary operators, the same proofs go through for flows as well.

######  Proposition 2.9.

The product of two ergodic flows is ergodic if and only if the flows have no
common eigenvalues.

Let $(\Gamma^{\mathbb{N}},\sigma)$ be a shift space. Given a function
$f:\Gamma^{\mathbb{N}}\rightarrow(0,+\infty)$, we may build a flow from
$(\Gamma^{\mathbb{N}},\sigma)$ by “flowing up” from a point
$\mathtt{i}\in\Gamma^{\mathbb{N}}$ until we reach the time $f(\mathtt{i})$,
then switch to the point $\sigma\mathtt{i}$ and continue flowing until the
time $f(\sigma\mathtt{i})$, and so on. Formally, we let
$Z=(\Gamma^{\mathbb{N}}\times\mathbb{R})/\sim$ eqipped with the quotinent
topology, where $\sim$ denotes the equivalence relation generated by
$(\mathtt{i},f(\mathtt{i}))\sim(\sigma\mathtt{i},0)$ for every
$\mathtt{i}\in\Gamma^{\mathbb{N}}$. Denoting each equivalence class
$[(\mathtt{i},t)]$ by the unique representative $(\mathtt{i},t)$ with $0\leq
t<f(\mathtt{i})$, we may write

$Z=\\{(\mathtt{i},t):\ \mathtt{i}\in\Gamma^{\mathbb{N}},0\leq t\leq
f(\mathtt{i})\\}.$

Let $T_{s}:(\mathtt{i},t)\mapsto(\mathtt{i},t+s)$ for every $s\geq 0$. If
there is a $\sigma$-invariant measure $\mu$ on $\Gamma^{\mathbb{N}}$ which
measures $f$, a natural $T_{s}$-invariant measure on $Z$ is given by
$\lambda:=(\mu\times\mathcal{L})_{Z}$. The tripet
$(\Gamma^{\mathbb{N}},T_{s},\lambda)$ is called the _suspension_ of
$(\Gamma^{\mathbb{N}},\sigma,\mu)$ under the roof function $f$. It is well-
known that if $(\Gamma^{\mathbb{N}},\sigma,\mu)$ is ergodic, then so is
$(Z,T_{s},\lambda)$.

A special property of regular enough suspension flows is that to any
eigenvalue corresponds a _continuous_ eigenfunction.

######  Proposition 2.10 ([24], Proposition 6.2).

Let $(Z,T_{s},\lambda)$ be the suspension of a shift space
$(\Gamma^{\mathbb{N}},\sigma,\mu)$ under a locally Hölder continuous roof
function, where $\mu$ is the equilibrium state for a locally Hölder continuous
potential on $\Gamma^{\mathbb{N}}$. Then a number $\alpha\in\mathbb{R}$ is an
eigenvalue of $(Z,T_{s},\lambda)$ for a continuous eigenfunction if and only
if it is an eigenvalue for a measurable eigenfunction.

It is well-known that Bernoulli measures on $\Gamma^{\mathbb{N}}$ are
equilibrium states for locally constant potentials.

######  Lemma 2.11.

Let $f:\Gamma^{\mathbb{Z}}\rightarrow(0,+\infty)$ be a continuous function
such that $f(\mathtt{i})=f(\mathtt{j})$ whenever
$\mathtt{i}^{+}=\mathtt{j}^{+}$, and let
$f^{+}:\Gamma^{\mathbb{N}}\rightarrow[0,+\infty)$ denote the function given by
$f^{+}(\mathtt{i})=f(\mathtt{j})$ for every $\mathtt{i}\in\Gamma^{\mathbb{N}}$
and $\mathtt{j}\in\Gamma^{\mathbb{Z}}$ such that $\mathtt{j}^{+}=\mathtt{i}$.

Let $\mu$ be a shift-invariant ergodic measure on $\Gamma^{\mathbb{Z}}$ which
measures $f$, and write $\mu^{+}$ for the projection onto the positive
coordinates. Let $(Z,T_{s},\lambda)=:\mathcal{Z}$ denote the suspension of
$(\Gamma^{\mathbb{Z}},\sigma,\mu)$ over $f$, and
$(Z^{+},T_{s},\lambda^{+})=:\mathcal{Z}^{+}$ the suspension of
$(\Gamma^{\mathbb{N}},\sigma,\mu^{+})$ over $f^{+}$.

The flows $\mathcal{Z}$ and $\mathcal{Z}^{+}$ have the same eigenvalues.

###### Proof.

Since $\mathcal{Z}$ factors onto $\mathcal{Z}^{+}$ through
$(\mathtt{i},t)\mapsto(\mathtt{i}^{+},t)$, it is not difficult to see that the
eigenvalues of $\mathcal{Z}^{+}$ are also eigenvalues of $\mathcal{Z}$.

For the other direction, suppose that $\alpha$ is not an eigenvalue of
$\mathcal{Z}^{+}$. Then by Lemma 2.7, the discrete-time system
$(Z^{+},T_{\alpha^{-1}},\lambda^{+})$ is ergodic. Let $U\subseteq Z$ be an
open set of the form $[\mathtt{i}]_{m}^{n}\times I$ for some
$\mathtt{i}\in\Gamma^{\mathbb{Z}}$, $m\leq n\in\mathbb{Z}$ and an interval
$I\subseteq\mathbb{R}$. Since any continuous function on $Z$ can be
approximated arbitrarily well (in $L^{1}$) by simple functions on this kind of
sets, in order to show that $\lambda$ is ergodic under $T_{1/\alpha}$, it
suffices to show that

$\lim_{n\rightarrow\infty}\frac{1}{n}\\#\\{0\leq k\leq n:\
T_{k/\alpha}(\mathtt{i},t)\in U\\}=\lambda(U)$

for a.e. $(\mathtt{i},t)$.

Let $\ell\in\mathbb{N}$ be large enough so that for every $(\mathtt{i},t)\in
T_{-\ell/\alpha}U$, also $(\mathtt{j},t)\in T_{-\ell/\alpha}U$ whenever
$\mathtt{j}^{+}=\mathtt{i}^{+}$. Write $(T_{-\ell/\alpha}U)^{+}$ for the
projection of $T_{-\ell/\alpha}U$ onto $Z^{+}$, and note that
$T_{-\ell/\alpha}U$ equals the embedding of $(T_{-\ell/\alpha}U)^{+}$ to $Z$.

Let $A^{+}$ be the set of full $\lambda^{+}$-measure such that for each
$(\mathtt{j},t)\in A^{+}$, we have

$\lim_{n\rightarrow\infty}\frac{1}{n}\\#\\{0\leq k\leq n:\
T_{k/\alpha}(\mathtt{j},t)\in(T_{-\ell/\alpha}U)^{+}\\}=\lambda^{+}((T_{-\ell/\alpha}U)^{+})=\lambda(T_{-\ell/\alpha}U)=\lambda(U).$

by Birkhoff’s ergodic theorem. The second-to-last equality follows from the
choice of $\ell$, and the last follows from $T_{t}$-invariance of $\lambda$.
Now, if $A$ is the embedding of $A^{+}$ to $Z$, then $T_{\ell/\alpha}A$ has
full $\lambda$-measure, and for each $(\mathtt{i},t)\in T_{\ell/\alpha}A$,

$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\\#\\{0\leq k\leq n:\
T_{k/\alpha}(\mathtt{i},t)\in U\\}$ $\displaystyle=\ $
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\\#\\{0\leq k\leq n:\
T_{(k-\ell)/\alpha}(\mathtt{i},t)\in T_{-\ell/\alpha}U\\}$ $\displaystyle=\ $
$\displaystyle\lambda(U).$

Therefore $\lambda$ is ergodic under $T_{1/\alpha}$ and by Lemma 2.7, $\alpha$
is not an eigenvalue of $\mathcal{Z}$. ∎

###  2.6. Shannon entropy

For a probability measure $\mu$ on $\mathbb{R}^{d}$ and any measurable
partitions $\mathcal{E}$ and $\mathcal{F}$ of $\mathbb{R}^{d}$, we write
$H(\mu,\mathcal{E})=-\sum_{E\in\mathcal{E}}\mu(E)\log\mu(E)$ for the
_(Shannon) entropy_ of $\mu$ with respect to $\mathcal{E}$, and
$H(\mu,\mathcal{E}|\mathcal{F})=-\sum_{F\in\mathcal{F}}\mu(F)H(\mu_{F},\mathcal{E})$
for the _conditional entropy_ of $\mu$ with respect to $\mathcal{E}$, given
$\mathcal{F}$.

Let $\mathcal{D}_{n}=\mathcal{D}_{n}(\mathbb{R}^{d})$ denote the partition of
$\mathbb{R}^{d}$ into dyadic cubes of side-length $2^{-n}$ which we call the
“level-$n$” dyadic partition. When $D\in\mathcal{D}_{n}$ and $\mu$ is a
measure with $\mu(D)>0$, let $\mu^{D}:=F\mu_{D}$, where $F$ is the homothety
sending $D$ onto $[-1,1)^{d}$. For $x\in\mathbb{R}^{d}$, write $D_{n}(x)$ for
the element of $\mathcal{D}_{n}$ that contains $x$. For entropy with respect
to the dyadic partition we use the short-hand notation
$H_{n}(\mu)=H(\mu,\mathcal{D}_{n})=-\sum_{D\in\mathcal{D}_{n}}\mu(D)\log\mu(D)$.
In the following, we record some elementary properties of entropy.

######  Lemma 2.12.

Let $\mu$ be a probability meausure on $\mathbb{R}^{d}$, and let $\mathcal{E}$
and $\mathcal{F}$ be partitions of $\mathbb{R}^{d}$ such that each element of
$\mathcal{E}$ intersects at most $k$ elements of $\mathcal{F}$ and vice versa.
Then

$|H(\mu,\mathcal{E})-H(\mu,\mathcal{F})|\leq O(k).$

######  Lemma 2.13 (Chain rule).

Let $\mu$ be a probability measure on $\mathbb{R}^{2}$, and let $\mathcal{E}$,
$\mathcal{F}$ be partitions of $\mathbb{R}^{2}$ such that $\mathcal{E}$
refines $\mathcal{F}$. Then

$H(\mu,\mathcal{E})=H(\mu,\mathcal{E}|\mathcal{F})+H(\mu,\mathcal{F}).$

Let $\mathcal{D}_{n}(\theta)$ denote the level-$n$ dyadic partition of
$\theta\in\mathbb{RP}^{1}$. The following is a simple application of the chain
rule; let $R_{\theta}$ denote the “shortest” rotation which takes
$\theta\in\mathbb{RP}^{1}$ onto the $y$-axis, with $R_{x\text{-axis}}$ given
by the clockwise rotation.

######  Lemma 2.14.

Let $\mu$ be a probability measure on $\mathbb{R}^{2}$, and let
$\theta\in\mathbb{RP}^{1}$. Then, denoting
$H_{n}(\mu|\pi_{\theta}):=H(\mu,R_{\theta^{\perp}}^{-1}\mathcal{D}_{n}(\mathbb{R}^{2})|\pi_{\theta}^{-1}\mathcal{D}_{n}(\theta))$,
we have

$|H_{n}(\mu)-(H_{n}(\pi_{\theta}\mu)+H_{n}(\mu|\pi_{\theta}))|\leq O(1)$

for every $n\in\mathbb{N}$.

######  Lemma 2.15 (Concavity and almost-convexity).

For any probability measures $\mu_{1},\ldots,\mu_{k}$ and a probability vector
$(p_{1},\ldots,p_{k})$,

$p_{i}H_{n}(\mu_{i})\leq H_{n}(\sum_{i=1}^{k}p_{i}\mu_{i})\leq
p_{i}H_{n}(\mu_{i})-\sum_{i=1}^{k}p_{i}\log p_{i}$

Taking a convolution can decrease entropy at most by an additive constant that
depends only on the dimension of the ambient space.

######  Lemma 2.16.

For any probability measures $\mu$ and $\nu$ and any $n$,

$H_{n}(\mu*\nu)\geq H_{n}(\mu)-O(1).$

###### Proof.

The claim follows by approximating $\nu$ in weak-∗ sense with a convex
combination of Dirac measures, and then applying Lemmas 2.15 and 2.12. ∎

######  Lemma 2.17.

Let $\mu$ and $\nu$ be probability measures on $[0,1]$, and suppose that $\mu$
is non-atomic. Then for every $r>0$ and $\varepsilon>0$, there exists
$N_{0}\in\mathbb{N}$ such that for any interval $I$ with $\mu(I)\geq r$, we
have

$\frac{1}{N}H_{N}(\mu_{I}*\nu)\geq\dim(\mu*\nu)-\varepsilon$

for every $N\geq N_{0}$.

###### Proof.

Let $K_{r}=\\{(a,b)\in\mathbb{R}^{2}:\ \mu([a,b])\geq r\\}$. We will show that
for every $r>0$ such that $K_{r}$ is nonempty,
$\frac{1}{N}H_{N}(\mu_{[a,b]}*\nu)$ is continuous in $(a,b)\in K_{r}$ (in the
subspace topology) and the continuity is uniform in $N$.

Let $r,\varepsilon>0$ be given, and let $\delta>0$ be small with respect to
$\varepsilon$ and $r$. Fix $(a_{0},b_{0})\in K_{r}$ and let $I_{\delta}$ be
the largest interval contained in $[a,b]$ for every $(a,b)\in
B((a_{0},b_{0}),\delta)$. If $\delta$ is small enough, we have
$\frac{\mu(I_{\delta})}{\mu([a,b])}\in[1-\varepsilon,1+\varepsilon]$ for every
$(a,b)\in B((a_{0},b_{0}),\delta)$, by non-atomicity of $\mu$ and the
assumption $\mu([a_{0},b_{0}])\geq r$.

Now, for every $(a,b),(a^{\prime},b^{\prime})\in B((a_{0},b_{0}),\delta)$,
applying bilinearity of convolution and Lemma 2.15 in the first and second-to-
last inequalities, we have

$\displaystyle H_{N}(\mu_{[a,b]}*\nu)$ $\displaystyle\leq\ $
$\displaystyle\frac{\mu(I_{\delta})}{\mu([a,b])}H_{N}(\mu_{I_{\delta}}*\nu)+\left(1-\frac{\mu(I_{\delta})}{\mu([a,b])}\right)H_{N}(\mu_{[a,b]\setminus
I_{\delta}}*\nu)+\varepsilon$ $\displaystyle\leq\ $
$\displaystyle(1+\varepsilon)\frac{\mu(I_{\delta})}{\mu([a^{\prime},b^{\prime}])}H_{N}(\mu_{I_{\delta}}*\nu)+2N\varepsilon$
$\displaystyle\leq\ $
$\displaystyle(1+\varepsilon)H_{N}(\mu_{[a^{\prime},b^{\prime}]}*\nu)+2N\varepsilon$
$\displaystyle\leq\ $ $\displaystyle
H_{N}(\mu_{[a^{\prime},b^{\prime}]}*\nu)+3N\varepsilon.$

Thus $(a,b)\mapsto\frac{1}{N}H_{N}(\mu_{[a,b]}*\nu)$ is continuous in $K_{r}$,
uniformly in $N$. On the other hand, for every $(a,b)\in K_{r}$,

$\dim(\mu*\nu)\leq\dim(\mu_{[a,b]}*\nu)\leq\liminf_{N\rightarrow\infty}\frac{1}{N}H_{N}(\mu_{[a,b]}*\nu)$

by Fatou’s lemma. By uniform continuity, this convergence is uniform in
$K_{r}$, which is what we wanted to prove. ∎

###  2.7. Magnifying measures

It turns out that the magnifications of a strictly self-affine measure along a
conformal partition have a very special structure. This structure makes it
much easier to analyse the convolutions of magnifications instead of the
convolution of self-affine measures directly, which in turn is put to use in
the proof of Theorem 1.1 through the local entropy averages. In the following,
we recap the standard terminology of magnifying measures, and some statistical
properties regarding sequences of magnifications.

For $x\in\mathbb{R}^{d}$ and $r\geq 0$, we let $T_{x}:\ y\mapsto y-x$ denote
the translation taking $x$ to the origin, and $S_{r}$ the exponential
“magnification” operation on measures, given for a measure $\mu$ whose support
contains the origin by

$S_{r}\mu(A)=\mu(B(0,2^{-r}))^{-1}\mu(2^{-r}A\cap B(0,2^{-r}))$

for every Borel $A\subseteq B(0,1)$. The following property of $S_{r}$-ergodic
measures appears also in [20].

######  Lemma 2.18.

Let $\mu$ be a Radon measure on $\mathbb{R}^{2}$, and let $P$ be a measure on
$\mathcal{P}(\mathbb{R}^{2})$ which is invariant and ergodic under
$(S_{r})_{r\geq 0}$. Then for any $\theta\in\mathbb{RP}^{1}$ such that
$\pi_{\theta}\mu$ is exact dimensional, we have

$\dim(\pi_{\theta}\nu*\pi_{\theta}\mu)=\min\\{1,\dim\pi_{\theta}\nu+\dim\pi_{\theta}\nu\\}$

for $P$-almost every $\nu$.

###### Proof.

For any probability measure $\nu$, it is not difficult to see that
$\pi_{\theta}S_{r}\nu\ll S_{r}\pi_{\theta}\nu$, hence
$\pi_{\theta}\mu*\pi_{\theta}S_{r}\nu\ll\pi_{\theta}\mu*S_{r}\pi_{\theta}\nu$
and
$\dim(\pi_{\theta}\mu*\pi_{\theta}S_{r}\nu)\geq\dim(\pi_{\theta}\mu*S_{r}\pi_{\theta}\nu)$.
Since the set of functions $\\{\mathbb{R}^{2}\rightarrow\mathbb{R},\
(x,y)\mapsto x+e^{r}y:\ r\geq 0\\}$ is a parametrization of orthogonal
projections onto lines with slope in $[1,\pi/2)$, the classical projection
theorem of Marstrand-Mattila asserts that

$\dim(\pi_{\theta}\mu*S_{r}\pi_{\theta}\nu)=\min\\{1,\dim\pi_{\theta}\mu+\dim\pi_{\theta}\nu\\}$

for $\mathcal{L}$-a.e. $r$. Since
$\dim\pi_{\theta}S_{r}\nu\geq\dim\pi_{\theta}S_{t}\nu$ for $r\geq t$ and
$P$-a.e. $\nu$, it follows from $S_{r}$-ergodicity of $P$ that
$\dim\pi_{\theta}\nu=:C$ is constant $P$-a.e.

Thus, using the $S_{r}$-invariance of $P$,

$\displaystyle\int\dim(\pi_{\theta}\mu*\pi_{\theta}\nu)\,dP(\nu)$
$\displaystyle=\int_{0}^{1}\int\dim(\pi_{\theta}\mu*\pi_{\theta}\nu)\,dS_{r}P(\nu)\,dr$
$\displaystyle=\int_{0}^{1}\int\dim(\pi_{\theta}\mu*\pi_{\theta}S_{r}\nu)\,dP(\nu)\,dr$
$\displaystyle\geq\int\int_{0}^{1}\dim(\pi_{\theta}\mu*S_{r}\pi_{\theta}\nu)\,dr\,dP(\nu)$
$\displaystyle=\int\min\\{1,\dim\pi_{\theta}\mu+\dim\pi_{\theta}\nu\\}\,dP(\nu)$
$\displaystyle=\min\\{1,\dim\pi_{\theta}\mu+C\\}.$

This implies that
$\dim(\pi_{\theta}\mu*\pi_{\theta}\nu)\geq\min\\{1,\dim\pi_{\theta}\mu+\dim\pi_{\theta}\nu\\}$
for $P$-a.e. $\nu$. The other direction is standard. ∎

There is a natural way in which measures on $\mathbb{R}^{d}$ give rise to
measures on $\mathcal{P}(\mathbb{R}^{d})$. Namely, consider the sequence
$(S_{r}T_{x}\mu)_{r\geq 0}$, called the _scenery_ of $\mu$ at $x$. The
statistical properties of this sequence are described by the accumulation
points of the sequence
$\left(\frac{1}{t}\int_{0}^{t}\delta_{S_{r}T_{x}\mu}\,dr\right)_{t\geq 1}$,
called the _scenery flow_ of $\mu$ at $x$. The accumulation points of the
scenery flow in the weak-∗ topology are measures on
$\mathcal{P}(\mathbb{R}^{d})$, and are called _tangent distributions_ of $\mu$
at $x$. The measure $\mu$ is called _uniformly scaling_ if the scenery flow
converges almost everywhere to a unique tangent distribution $P$. It is then
said that $\mu$ generates $P$.

A remarkable result of Hochman [15] is that tangent distributions at almost
every point are _fractal distributions_ , objects which enjoy strong spatial
invariance properties. We will give the definition here for completeness,
although we will not use it directly.

######  Definition 2.19.

An $S_{r}$-invariant measure $P$ on $\mathcal{P}(\mathbb{R}^{d})$ is called a
_fractal distribution_ if for any measurable $A$, $P(A)=1$ if and only if for
every $r>0$, $P$-almost every $\nu$ satisfies

$S_{r}T_{x}\nu\in A$

for $\nu$-almost every $x$ with $B(x,e^{-r})\subseteq B(0,1)$.

######  Theorem 2.20 (Theorem 1.7, [15]).

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Then for $\mu$-almost every
$x$, every tangent distribution at $x$ is a fractal distribution.

######  Lemma 2.21.

Let $P$ be a fractal distribution. Then for $P$-a.e. $\nu$, any line $L$ with
$\nu(L)>0$ must contain the origin.

###### Proof.

We first show that $P$-a.e. atomic measure is the point mass at the origin.
This is almost immediate from the results of [15].

For a contradiction, let $A=\\{\nu:\ \nu(\\{x\\})>0,\ x\neq 0\\}$ and suppose
that $P(A)>0$. Let $P^{\prime}$ be an ergodic component of $P$ with
$P^{\prime}(A)>0$, and let $\eta\in A$ be a uniformly scaling measure
generating $P^{\prime}$. Indeed, it is not difficult to verify from the
definition of a fractal distribution that $P^{\prime}$-almost every measure is
uniformly scaling. Let $x$ be such that $\eta(\\{x\\})>0$. Since
$\eta|_{\\{x\\}}\ll\eta$, also $\eta|_{\\{x\\}}$ generates $P^{\prime}$ by an
application of the Lebesgue-Besicovitch differentiation theorem, whence
$P^{\prime}$ is supported on the point mass at the origin. In particular,
$P^{\prime}(A)=0$, a contradiction.

Now, to prove the statement of the lemma, suppose that there exists a set $B$
with $P(B)>0$ such that for every $\nu\in B$, there exists a line $L_{\nu}$
with $\nu(L_{\nu})>0$ and $0\not\in L_{\nu}$. Let $P^{\prime}$ be an ergodic
component of $P$ with $P^{\prime}(B)>0$, and let $\eta\in B$ be a uniformly
scaling measure generating $P^{\prime}$. Now, since $\eta(L_{\eta})>0$, also
the measure $\eta|_{L_{\eta}}$ generates $P^{\prime}$. Let $L$ denote the line
$L_{\eta}$ translated so that it contains the origin. Clearly, all tangent
measures of $\eta|_{L_{\eta}}$ are supported on $L$, whence $P^{\prime}$ is
supported on measures which are supported on $L$. Since any line not
containing the origin intersects $L$ in at most one point, such a line has
$P^{\prime}$-almost surely zero measure by the above. Thus $P^{\prime}(B)=0$
which is a contradiction. ∎

We can also choose to magnify $\mu$ along a discrete sequence of scales. For
$N\in\mathbb{N}$, we call the sequence
$\left(\frac{1}{nN}\sum_{k=1}^{n}\delta_{S_{kN}T_{x}\mu}\right)_{n\in\mathbb{N}}$
the _$N$ -scenery process_ of $\mu$ at $x$. Similarly, weak-∗ accumulation
points of this sequence we call $N$_-tangent distributions_. While these are
a-priori different objects from the accumulation points of the continuous-time
scenery flow, most properties that are typical for tangent distributions are
also typical for $N$-tangent distributions.

######  Lemma 2.22.

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. For $\mu$-almost every $x$,
any $N\in\mathbb{R}$ and any $N$-tangent distribution $P$ of $\mu$ at $x$, the
following holds: For $P$-almost every $\nu$, any line $L$ with $\nu(L)>0$ that
intersects the interior of $B(0,1)$ must contain the origin.

###### Proof.

It follows from [15, Proposition 5.5] that for $\mu$-almost every $x$, if $P$
is an $N$-tangent distribution at $x$, then the measure

$Q:=\int\int_{0}^{N}\delta_{S_{r}\nu}\,dr\,dP(\nu)$

is a fractal distribution. Therefore by Lemma 2.21, for $P$-a.e. $\nu$ and
$\mathcal{L}$-almost every $r\in[0,N]$, any line $L$ with $S_{r}\nu(L)>0$ must
contain the origin. Taking $r\rightarrow 0$ along a countable sequence, we see
that for $P$-almost every $\nu$, any line $L$ with $\nu(L)>0$ that intersects
the interior of $B(0,1)$ must contain the origin. ∎

###  2.8. Conditional measures on lines

For a measure $\mu$ on $\mathbb{R}^{2}$ and $\theta\in\mathbb{RP}^{1}$, let
$\mu=\int\mu_{x}^{\theta}\,d\pi_{\theta}\mu(x)$ denote the disintegration of
$\mu$ with respect to $\pi_{\theta}$. It is well-known that for almost every
$x\in\theta$, the measure $\mu_{x}^{\theta}$ is supported on the line
$x+\theta^{\perp}$, and that $\mu_{x}^{\theta}$ is the limit of normalized
restrictions of $\mu$ on thinner and thinner tubes centered at
$x+\theta^{\perp}$. It will be useful for us to know that these tubes can
replaced by preimages of sets of relatively large measure.

######  Lemma 2.23.

Let $\mu$ be a measure on $\mathbb{R}^{2}$, let $\delta>0$ and
$\theta\in\mathbb{RP}^{1}$. For every $r>0$, let

$\mathcal{I}(x,r,\delta)=\\{I\subseteq B(x,r):\
\frac{\pi_{\theta}\mu(I)}{\pi_{\theta}\mu(B(x,r))}\geq\delta\\}.$

Then for $\pi_{\theta}\mu$-almost every $x$, we have

$\lim_{r\rightarrow 0}\sup_{I\in\mathcal{I}(x,r,\delta)}d_{\rm
LP}\left(\mu_{\pi_{\theta}^{-1}(I)},\ \mu_{x}^{\theta}\right)=0.$

###### Proof.

Let $\varepsilon>0$, and let $E_{n}\subseteq\theta$ be the compact set given
by Lusin’s theorem with $\pi_{\theta}\mu(E_{n})\geq 1-1/n$, on which the
function $y\mapsto\mu_{y}^{\theta}$ is continuous. Since
$\pi_{\theta}\mu(\bigcup_{n\in\mathbb{N}}E_{n})=1$, it suffices to prove the
statement for almost every $x\in E_{n}$, for every $n\in\mathbb{N}$. Now, for
almost every $x\in E_{n}$, if $B:=B(x,r)$ and $I\in\mathcal{I}(x,r,\delta)$,

$\displaystyle\frac{\pi_{\theta}\mu(I\cap
E_{\varepsilon})}{\pi_{\theta}\mu(I)}$
$\displaystyle=1-\frac{\pi_{\theta}\mu(I\setminus
E_{\varepsilon})}{\pi_{\theta}\mu(I)}$ $\displaystyle\geq
1-\frac{\pi_{\theta}\mu(B\setminus
E_{\varepsilon})}{\pi_{\theta}\mu(B)}\frac{\pi_{\theta}\mu(B)}{\pi_{\theta}\mu(I)}$
$\displaystyle\geq 1-\frac{1}{\delta}\frac{\pi_{\theta}\mu(B\setminus
E_{\varepsilon})}{\pi_{\theta}\mu(B)}$ $\displaystyle=1-o(r)/\delta$

by an application of the Lebesgue-Besicovitch differentiation theorem.

Therefore, for any Borel set $A\subseteq\mathbb{R}^{2}$,
$I\in\mathcal{I}(x,r,\delta)$ and $\varepsilon>0$,

$\displaystyle\mu_{\pi_{\theta}^{-1}I}(A^{\varepsilon})$
$\displaystyle=\frac{1}{\pi_{\theta}\mu(I)}\mu(\pi_{\theta}^{-1}I\cap
A^{\varepsilon})$
$\displaystyle=\frac{1}{\pi_{\theta}\mu(I)}\int_{I}\mu_{y}^{\theta}(A^{\varepsilon})\,d\pi_{\theta}\mu(y)$
$\displaystyle\geq\frac{1-o(r)/\delta}{\pi_{\theta}\mu(I\cap
E_{\varepsilon})}\int_{I\cap
E_{n}}\mu_{y}^{\theta}(A^{\varepsilon})\,d\pi_{\theta}\mu(y)$
$\displaystyle\geq(1-o(r)/\delta)(\mu_{x}^{\theta}(A)-\varepsilon)$

if $r$ is small enough, by continuity of $y\mapsto\mu_{y}^{\theta}$.
Similarly,

$\mu_{\pi_{\theta}^{-1}I}(A)\leq\frac{1}{\pi_{\theta}\mu(I\cap
E_{n})}\int_{I\cap
E_{\varepsilon}}\mu_{y}^{\theta}(A^{\varepsilon})\,d\pi_{\theta}\mu(y)+o(r)/\delta\leq\mu_{x}^{\theta}(A^{\varepsilon})+o(r)/\delta+\varepsilon,$

so $d_{\rm LP}(\mu_{\pi^{-1}I},\ \mu_{x}^{\theta})\leq 2\varepsilon$ for small
enough $r$. ∎

##  3\. On local entropy averages

In this section, we recall the local entropy averages of [19] and introduce
the different notions of magnifactions of measures that we use. Let $\mu$ and
$\nu$ be self-affine measures as in the statement of Theorem 1.1, and denote
by $\bar{\mu}$, $\bar{\nu}$ the associated Bernoulli measures.

For real numbers $a$ and $b$, write $a\approx b$ to indicate that they are
multiplicatively comparable. By Lemmas 2.1, 2.3 and basic geometry,

$\|B_{\mathtt{j}|_{n}}\|\approx|\pi_{\theta(\mathtt{j})}B_{\mathtt{j}|_{n}}B(0,1)|\approx|\pi_{\theta^{*}(\mathtt{j})}B_{\mathtt{j}|_{n}}B(0,1)|$

for all $\mathtt{j}\in\Lambda^{\mathbb{N}}$ and $n\in\mathbb{N}$, where the
constant of comparability depends only on the IFS $\Psi$. In order to simplify
notation, suppose that
$1/2\leq\|B_{\mathtt{j}|_{n}}\|^{-1}|\pi_{\theta^{*}(\mathtt{j})}B_{\mathtt{j}|_{n}}B(0,1)|\leq
2$ for every $\mathtt{j}$ and $n$.

For every $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and
$\mathtt{j}\in\Lambda^{\mathbb{N}}$, we define the “stopping times”

$\displaystyle i_{k}=i_{k}(\mathtt{i})=\min\\{n\in\mathbb{N}:\
|\pi_{\theta^{*}(\mathtt{i})}A_{\mathtt{i}|_{n}}B(0,1)|\leq 2^{-kN}\\},$
$\displaystyle i_{k}=i_{k}(\mathtt{j})=\min\\{n\in\mathbb{N}:\
|\pi_{\theta^{*}(\mathtt{j})}B_{\mathtt{j}|_{n}}B(0,1)|\leq 2^{-kN}\\}.$

Although the stopping times on $\Gamma^{\mathbb{N}}$ and
$\Lambda^{\mathbb{N}}$ are denoted by the same letter $i_{k}$, the choice of
domain will always be clear from the context: for example,
$\mathtt{i}|_{i_{k}}:=\mathtt{i}|_{i_{k}(\mathtt{i})}$ and
$\mathtt{j}|_{i_{k}}:=\mathtt{j}|_{i_{k}(\mathtt{j})}$. Then for every $k$,

$\displaystyle\|A_{\mathtt{i}|_{i_{k}}}\|\approx\|B_{\mathtt{j}|_{i_{k}}}\|\approx
2^{-kN}.$

We now define the different notions of scale-$k$ magnifications of $\mu$ and
$\nu$.

######  Notation 3.1.

For each $\mathtt{i}\in\Gamma^{\mathbb{N}}$,
$\mathtt{j}\in\Lambda^{\mathbb{N}}$ and $k\in\mathbb{N}$, let

$\displaystyle\nu_{\mathtt{j}|_{i_{k}}}$
$\displaystyle:=S_{kN}T_{\Pi(\sigma^{i_{k}}\mathtt{j})}\psi_{\mathtt{j}|_{i_{k}}}\nu,$
$\displaystyle\mu_{\mathtt{i}|_{i_{k}}}$
$\displaystyle:=S_{kN}T_{\Pi(\sigma^{i_{k}}\mathtt{i})}\varphi_{\mathtt{i}|_{i_{k}}}\nu,$
$\displaystyle\mu^{\mathtt{i},k}$
$\displaystyle:=\mu^{D_{kN}(\Pi(\mathtt{i}))}.$

Note that $\mu_{\mathtt{i}|_{i_{k}}}$ and $\nu_{\mathtt{j}|_{i_{k}}}$ are
measures supported on a thin ellipses whose major axes have length comparable
to $1$ and are oriented in the directions $\theta(A_{\mathtt{i}|_{i_{k}}})$
and $\theta(B_{\mathtt{j}|_{i_{k}}})$, respectively. On the other hand,
$\mu^{\mathtt{i},k}$ is a measure supported on $[-1,1]^{2}$, the “dyadic
magnification” of $\mu$.

We require the following form of the local entropy averages of [19]:

######  Theorem 3.2.

For any $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that if

$\liminf_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{N}H_{N}(\mu^{\mathtt{i},k}*\nu_{\mathtt{j}|_{i_{k}}})\geq
c\ \ \text{or}\ \
\liminf_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{N}H_{N}(\mu_{\mathtt{i}|_{i_{k}}}*\nu_{\mathtt{j}|_{i_{k}}})\geq
c$

for $\bar{\mu}\times\bar{\nu}$-a.e. $(\mathtt{i},\mathtt{j})$, then
$\dim(\mu*\nu)\geq c-\varepsilon$.

The proof is essentially in [19], but we provide a sketch for the proof of the
first half of the statement for the convenience of the reader. The second half
goes through similarly. We begin by recalling some terminology of [19]. In the
following, let $X$ and $Y$ be finite sets, let $0<\rho<1$, and let $d$ be a
metric on $X^{\mathbb{N}}$ such that
$d(\mathtt{x},\mathtt{x^{\prime}})\approx\rho^{|\mathtt{x}\wedge\mathtt{x}^{\prime}|}$.
Following [19], we say that

  * i)

a map $g:X^{\mathbb{N}}\rightarrow Y^{\mathbb{N}}$ is a _tree morphism_ if,
for every $n\in\mathbb{N}$ and length-$n$ cylinder $[x_{1}\ldots
x_{n}]\subseteq X^{\mathbb{N}}$ there exists a length-$n$ cylinder
$[y_{1}\ldots y_{n}]\subseteq Y^{\mathbb{N}}$ such that $g([x_{1}\ldots
x_{n}])\subseteq[y_{1}\ldots y_{n}]$,

  * ii)

a map $h:X^{\mathbb{N}}\rightarrow\mathbb{R}^{d}$ is _faithful_ if there
exists a constant $C$ such that for any $[x_{1}\ldots x_{n}]\subseteq
X^{\mathbb{N}}$, no point in $f([x_{1}\ldots x_{n}])$ is covered by more than
$C$ of the sets $f([x_{1}\ldots x_{n}x])$ for $x\in X$, and $f([x_{1}\ldots
x_{n}])$ contains a ball of radius $(C^{-1}\rho)^{n}$ and is contained in a
ball of radius $(C\rho)^{n}$.

###### Proof of Theorem 3.2.

Let $N\in\mathbb{N}$. As in [19], the idea is to lift $\mu\times\nu$ to a
measure $\eta$ on the tree

$\Sigma:=\\{1,\ldots,2^{N}\\}^{\mathbb{N}}\times\Lambda^{\mathbb{N}},$

and then find a tree morphism $g_{N}:\Sigma\rightarrow Y_{N}^{\mathbb{N}}$ and
a faithful map $h_{N}:Y_{N}^{\mathbb{N}}\rightarrow\mathbb{R}^{2}$ such that
$(h_{N}\circ g_{N})\eta=\mu*\nu$.

On $\Sigma$, let distance between pairs $(\mathtt{k},\mathtt{j})$ and
$(\mathtt{k}^{\prime},\mathtt{j}^{\prime})$ be defined as the number

$\max\\{2^{-N\cdot\max\\{n:\
\mathtt{k}|_{n}=\mathtt{k}^{\prime}|_{n}\\}},2^{-N\cdot\max\\{n:\
\mathtt{j}|_{i_{n}}=\mathtt{j}^{\prime}|_{i_{n}}\\}}\\}.$

Then the topology on $\Sigma$ is generated by sets of the form
$[\mathtt{k}|_{n}]\times[\mathtt{j}|_{i_{n}}]$,
$\mathtt{k}\in\\{1,\ldots,2^{N}\\}^{\mathbb{N}}$,
$\mathtt{j}\in\Lambda^{\mathbb{N}}$.

Write $D_{2^{N}}(\mathbb{R}^{2})=\\{D_{1},\ldots,D_{2^{N}}\\}$ and let $F_{k}$
denote the map which sends $[-1,1]$ onto the closure of $D_{k}$. Let $\Pi_{1}$
denote the map
$\\{1,\ldots,2^{N}\\}\rightarrow[-1,1]^{2},\mathtt{k}\mapsto\lim_{n\rightarrow\infty}F_{\mathtt{k}|_{n}}(0)$,
and $\Pi_{2}$ the map
$\Lambda^{\mathbb{N}}\rightarrow\mathbb{R}^{2},\mathtt{j}\mapsto\lim_{k\rightarrow\infty}\psi_{\mathtt{j}|_{k}}(0)$.
Using the maps $\Pi_{1}$ and $\Pi_{2}$, the measure $\mu\times\nu$ can be
lifted to a measure $\eta$ on $\Sigma$ (by possibly translating the dyadic
partition so that the boundaries of dyadic squares have $\mu$-measure $0$).
Let $\Pi_{0}$ denote the map
$\Sigma\rightarrow\mathbb{R}^{2}\times\mathbb{R}^{2},(\mathtt{k},\mathtt{j})\mapsto(\Pi_{1}(\mathtt{k}),\Pi_{2}(\mathtt{j}))$
and $*$ the map
$\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},(x,y)\mapsto x+y$
so that $\Pi_{0}\eta=\mu\times\nu$ and $(*\circ\Pi)\eta=\mu*\nu$.

We will now construct a tree $Y_{N}^{\mathbb{N}}$, a tree morphism
$g_{N}:\Sigma\rightarrow Y_{N}^{\mathbb{N}}$ and a faithful map
$h_{N}:Y_{N}^{\mathbb{N}}\rightarrow\mathbb{R}^{2}$ such that the following
diagram commutes:

${\Sigma}$${Y_{N}^{\mathbb{N}}}$${\mathbb{R}^{2}}$$\scriptstyle{g_{N}}$$\scriptstyle{*\circ\Pi}$$\scriptstyle{h_{N}}$

Let $Y_{N}=\\{(\frac{k}{2^{N}},\frac{\ell}{2^{N}}):\ 0\leq k,\ell\leq
2^{N}-1\\}$, and associate to each $x\in Y_{N}$ the square $Q_{x}$ of side
length $2^{-N+4}$ and bottom left corner at $x$. This gives an overlapping
cover for $[-1,1]^{2}$. Writing $S_{x}$ for the affine contraction which sends
$[-1,1]^{2}$ to $Q_{x}$, there is a faithful surjection
$h_{N}:Y_{N}^{\mathbb{N}}\rightarrow[-1,1]^{2}$,
$h_{N}(x_{1}x_{2}x_{3}\ldots)=\lim_{n\rightarrow\infty}S_{x_{1}}\circ\cdots\circ
S_{x_{n}}(0)$. Let the metric on $Y_{N}^{\mathbb{N}}$ be given by
$d(\mathtt{x},\mathtt{y})=2^{-N|\mathtt{x}\wedge\mathtt{y}|}$ for
$\mathtt{x},\mathtt{y}\in Y_{N}^{\mathbb{N}}$

We construct $g_{N}$ iteratively. First observe that for any
$[\mathtt{k}|_{n}]\times[\mathtt{j}|_{i_{n}}]$, we have
$\Pi_{1}[\mathtt{k}|_{1}]+\Pi_{2}[\mathtt{j}|_{i_{1}}]\subseteq Q_{a_{1}}$ for
some $Q_{a_{1}}$, since $\|B_{\mathtt{j}|_{1}}\|\leq
2|\pi_{\theta(\mathtt{j})^{*}}B_{\mathtt{j}|_{1}}|\leq 2^{-N+1}$ and thus
$\operatorname{diam}(\Pi_{1}[\mathtt{k}|_{1}]+\Pi_{2}[\mathtt{j}|_{i_{1}}])\leq
2^{-N+2}$. Supposing that the sequence $a_{1}\ldots a_{n}$ has been
determined, we choose $a_{n+1}$ so that
$\Pi_{1}[\mathtt{k}|_{n+1}]+\Pi_{2}[\mathtt{j}|_{i_{n+1}}]\subseteq
Q_{a_{1}\ldots a_{n}a_{n+1}}$. Since $\operatorname{diam}(Q_{a_{1}\ldots
a_{n}})\rightarrow 0$, we may define
$g_{N}(\mathtt{k},\mathtt{j})=a_{1}a_{2}\ldots$. It is evident from the
construction that
$g_{N}([\mathtt{k}|_{n}]\times[\mathtt{j}|_{i_{n}}])\subseteq[a_{1}a_{2}\ldots
a_{n}]$, that is, sets of the form
$[\mathtt{k}|_{n}]\times[\mathtt{j}|_{i_{n}}]$ are mapped into cylinders of
$Y_{N}^{\mathbb{N}}$ by $g_{N}$. While this property of $g_{N}$ is strictly
speaking weaker than that of the tree morphism defined earlier, it is enough
for our purposes. By construction, we have $h_{N}\circ g_{N}=*\circ\Pi$.

Since $h_{N}$ is faithful, [19, Proposition 5.3] asserts that it preserves
entropy in the sense that

$|H_{(k+1)N}(g_{N}\eta_{[\mathtt{k}|_{k}]\times[\mathtt{j}|_{i_{k}}]})-H_{(k+1)N}(\mu_{\Pi_{1}[\mathtt{k}|_{k}]}*\psi_{\mathtt{j}|_{i_{k}}}\nu)|\leq
O(1)$

for every $k\in\mathbb{N}$. Since rescaling a measure does not change its
entropy much if we change the scale of the entropy by the same amount, by
Lemma 2.12, we have

$|H_{(k+1)N}(\mu_{\Pi_{1}[\mathtt{k}|_{k}]}*\psi_{\mathtt{j}|_{i_{k}}}\nu)-H_{N}(\mu^{\mathtt{i},k}*\nu_{\mathtt{j}|_{i_{k}}})|\leq
O(1)$

where $\mathtt{i}\in\Gamma^{\mathbb{N}}$ is such that
$\Pi(\mathtt{i})=\Pi_{1}(\mathtt{k})$. Therefore, by the assumption and the
local entropy averages for measures on trees [19, Theorem 4.4], we have $\dim
g_{N}\eta\geq c$. We remark here that in the hypothesis of [19, Theorem 4.4]
it is assumed that $g_{N}$ is a tree morphism, but it is not difficult to
verify that just having the property for sets of the form
$[\mathtt{k}|_{n}]\times[\mathtt{j}|_{i_{n}}]$ is enough to apply the theorem
in our setting. Finally, taking $g_{N}\eta$ through the faithful map $h_{N}$
yields $\mu*\nu$ and distorts the dimension by at most $O(1/N)$, by [19,
Proposition 5.2], which completes the proof. ∎

##  4\. Proof of Theorem 1.1

In this section, we state our key technical results, and apply them to prove
Theorem 1.1. We begin with some notation.

Recall that we defined $\mathbb{RP}^{1}$ as the collection of one-dimensional
subspaces of $\mathbb{R}^{2}$. Through the identification
$\mathbb{RP}^{1}\cong[0,\pi)$ we use $\theta\in[0,\pi)$ to denote both angles
and lines (making that angle with the positive $x$-axis). Recall that
$R_{\theta}$ denotes the “shortest” rotation which takes $\theta$ onto
$0^{\perp}$ (the $y$-axis), and for $\theta=0$, we choose the clockwise
rotation. For almost every $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and
$\theta\in\mathbb{RP}^{1}$, we let $\mu_{\mathtt{i},\theta}$ denote the
probability measure supported on the $y$-axis, obtained by taking the
conditional measure $\mu_{\Pi(\mathtt{i})}^{\theta}$ from the disintegration
$\mu=\int\mu_{x}^{\theta}\,d\pi_{\theta}\mu=\int\mu_{\Pi(\mathtt{i})}^{\theta}\,d\bar{\mu}$
supported on the line $\theta^{\perp}+\Pi(\mathtt{i})$, translating it by
$T_{\Pi(\mathtt{i})}$ and finally rotating it by $R_{\theta}$. That is,

(4.1)
$\mu_{\mathtt{i},\theta}=R_{\theta}T_{\Pi(\mathtt{i})}\mu_{\Pi(\mathtt{i})}^{\theta}.$

For $t\geq 0$, write $\mu_{\mathtt{i},\theta,t}:=S_{t}\mu_{\mathtt{i},t}$. The
measures $\mu_{\mathtt{i},\theta}$ are occasionally called _slices_ of $\mu$.

Fix a large integer $N$, and for $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and
$k\in\mathbb{N}$, let $\ell_{k}=\ell_{k}(\mathtt{i})$ be any increasing
sequence such that $\lim_{k\rightarrow\infty}\ell_{k}=\infty$ and

(4.2)
$B(\Pi(\mathtt{i}),2^{-kN})\cap\Pi(\Gamma^{\mathbb{N}})=B(\Pi(\mathtt{i}),2^{-kN})\cap\varphi_{\mathtt{i}|_{\ell_{k}}}(B(0,1))$

for every $k$. Such a sequence exists for every $\mathtt{i}$ by the strong
separation condition.

Our main geometric observation is that for any self-affine measure $\mu$ with
$\dim\mu>1$, the measures $\mu^{\mathtt{i},k}$ have a fiber structure in the
sense that $\pi_{\theta(\mathtt{i})^{\perp}}\mu^{\mathtt{i},k}$ is very close
to a slice of the original measure $\mu$, in a direction typical to the
Furstenberg measure $\mu_{F}$. This is true also when $\dim\mu\leq 1$, but in
this case the proof is slightly different. Throughout the paper, we adopt the
convention that $-\mu$ denotes the push-forward of $\mu$ through the map
$x\mapsto-x$. Recall that $\pi^{2}$ denotes the orthogonal projection onto the
$y$-axis.

######  Proposition 4.1 (Fiber structure).

Let $\mu$ be as in Theorem 1.1 with $\dim\mu>1$, and suppose that the
domination condition is satisfied. For any $\varepsilon>0$, the following
holds after an affine change of coordinates:

For $\bar{\mu}$-almost every $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and all
$\theta$ in a set of positive $\mu_{F}$-measure, there exists
$m\in\mathbb{N}$, a sequence of intervals $(I_{k})_{k}$ of length $2^{-m}$ and
a set $\mathcal{N}_{\varepsilon}\subseteq\mathbb{N}$ with
$\liminf_{n\rightarrow\infty}\frac{\\#(\mathcal{N}_{\varepsilon}\cap[0,n])}{n}\geq
1-\varepsilon$ such that

$d_{\rm LP}(\pi^{2}\mu^{\mathtt{i},k+m},\
\left(\rho(\ell_{k},(\mathtt{i},\theta))\mu_{M^{\ell_{k}}(\mathtt{i},\theta),kN+\log\alpha_{1}(\mathtt{i}|_{\ell_{k}})}\right)^{I_{k}})<\varepsilon$

for every $k\in\mathcal{N}_{\varepsilon}$, where
$\rho:\mathbb{N}\times\Gamma^{\mathbb{N}}\times\mathbb{RP}^{1}\rightarrow\\{-1,1\\}$
is a cocycle.

The domination assumption can be removed from the proposition at the cost of
having the change of coordinates depend on the word
$\mathtt{i}\in\Gamma^{\mathbb{N}}$; Furthermore, we leave it for the
interested reader to verify that if $\pi^{2}$ is replaced by
$\pi_{\theta(\mathtt{i})^{\perp}}$ and one chooses to magnify along balls
centered at $\Pi(\mathtt{i})$ instead of dyadic squares, then no domination
condition, change of coordinates nor restriction on the intervals
$(I_{k})_{k}$ is required. Such a modification is not required in the proof of
Theorem 1.1, however. The proof of the proposition is given in Section 5.

Let us briefly compare Proposition 4.1 with the existing results on the
scenery of self-affine measures. First, the result of Ferguson et. al. [13]
deals with measures on Bedford-McMullen carpets. In this case, the Furstenberg
measure $\mu_{F}$ equals a point mass, and the defining matrices involve no
reflections, whence the cocycle $\rho$ above may be replaced by the identity
and the map $M$ is just the left shift on the first argument. Using the carpet
structure, the authors also describe the fibers of $\mu^{\mathtt{i},k}$ with
respect to $\pi^{2}$, which seems to be difficult in our setting.

The result of Kempton [22], on the other hand, deals with self-affine measures
under the irreducibility and domination conditions. In addition, the methods
of [22] assume that the orthogonal projection of $\mu$ in $\mu_{F}$-almost
every direction is absolutely continuous, and that the defining affine
contractions preserve orientation. The projection condition allows Kempton to
deduce that each fiber of $\mu^{\mathtt{i},k}$ with respect to $\pi^{2}$ is
just the Lebesgue measure, while assuming the defining contractions to be
orientation-preserving allows the cocycle $\rho$ to be replaced with the
identity. Finally, Proposition 4.1 considers magnifications of $\mu$ along the
dyadic squares instead of balls centered at $\Pi(\mathtt{i})$, which brings in
some additional technical considerations regarding the distribution of $\mu$
near the boundaries of dyadic squares.

###  4.1. Estimating the local entropy averages

Recall from Section 3 that we have to find a lower bound for either
$\frac{1}{N}H_{N}(\mu^{\mathtt{i},k}*\nu_{\mathtt{j}|_{i_{k}}})$ or
$\frac{1}{N}H_{N}(\mu_{\mathtt{i}|_{i_{k}}}*\nu_{\mathtt{j}|_{i_{k}}})$, for
most $k$. Bounding the second quantity is easier, but the bound we obtain is
only efficient when $\dim\mu\geq\dim\nu\geq 1$ or $1\geq\dim\mu\geq\dim\nu$:

######  Claim 4.2.

Let $\mu$ and $\nu$ be as in Theorem 1.1. For $\bar{\mu}$-a.e.
$\mathtt{i}\in\Gamma^{\mathbb{N}}$, $\bar{\nu}$-a.e.
$\mathtt{j}\in\Lambda^{\mathbb{N}}$ and every $\varepsilon>0$,

$\liminf_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{N}H_{N}(\mu_{\mathtt{i}|_{i_{k}}}*\nu_{\mathtt{j}|_{i_{k}}})\geq\min\\{1,\dim\mu\\}+\min\\{1,\dim\nu\\}-\varepsilon$

for all large enough $N$.

The proof is given in Section 6. Applying Theorem 3.2, we get that

$\dim(\mu*\nu)\geq\min\\{1,\dim\mu\\}+\min\\{1,\dim\nu\\}.$

From this it readily follows that if

$\dim(\mu*\nu)<\min\\{2,\dim\mu+\dim\nu\\},$

then $\dim\mu>1>\dim\nu$. This was one of the statements of Theorem 1.1. From
now on, we suppose that $\dim\mu>1>\dim\nu$, and that $\Phi$ and $\Psi$
satisfy the domination condition.

In this case, bounding the quantity
$\frac{1}{N}H_{N}(\mu^{\mathtt{i},k}*\nu_{\mathtt{j}|_{i_{k}}})$ is more
efficient. Applying Lemmas 2.13 and 2.16 we see that

(4.3)
$\displaystyle\frac{1}{N}H_{N}(\mu^{\mathtt{i},k}*\nu_{\mathtt{j}|_{i_{k}}})$
$\displaystyle\geq\ $
$\displaystyle\frac{1}{N}H_{N}(\pi^{2}\mu^{\mathtt{i},k}*\pi^{2}\nu_{\mathtt{j}|_{i_{k}}})+\frac{1}{N}H_{N}(\mu^{\mathtt{i},k}|\pi^{2})-O(1/N)$
$\displaystyle=\ $
$\displaystyle\frac{1}{N}H_{N}(\pi^{2}\mu^{\mathtt{i},k}*\pi^{2}\nu_{\mathtt{j}|_{i_{k}}})+\frac{1}{N}H_{N}(\mu^{\mathtt{i},k})-\frac{1}{N}H_{N}(\pi^{2}\mu^{\mathtt{i},k})-O(1/N)$

for all $N$. Here, the average of the terms
$\frac{1}{N}H_{N}(\mu^{\mathtt{i},k})$ over $k$ is close to $\dim\mu$. Thus,
Theorem 1.1 follows from the following inequalities together with the local
entropy averages:

######  Claim 4.3.

Let $\mu$ be as in Theorem 1.1. For any $\varepsilon>0$ and large enough $N$,
the following holds after the affine change of coordinates of Proposition 4.1:
For $\bar{\mu}\times\mu_{F}$-a.e. $(\mathtt{i},\theta)$, we have

$\limsup_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{N}H_{N}(\pi^{2}\mu^{\mathtt{i},k})\leq\dim\mu-1+\varepsilon.$

######  Claim 4.4.

Let $\mu$ and $\nu$ be as in Theorem 1.1. Suppose that for some
$(i,j)\in\Gamma\times\Lambda$,
$\frac{\log|\lambda_{1}(A_{i})|}{\log|\lambda_{2}(B_{j})|}\not\in\mathbb{Q}$.
Then for any $\varepsilon>0$ and large enough $N$, the following holds after
the affine change of coordinates of Proposition 4.1: For $\bar{\mu}$-a.e.
$\mathtt{i}\in\Gamma^{\mathbb{N}}$ and $\bar{\nu}$-a.e.
$\mathtt{j}\in\Lambda^{\mathbb{N}}$, we have

$\liminf_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{N}H_{N}(\pi^{2}\mu^{\mathtt{i},k}*\pi^{2}\nu_{\mathtt{j}|_{i_{k}}})\geq\min\\{1,\dim\mu-1+\dim\nu\\}-\varepsilon.$

These claims are also proved in Section 6. We show how to conclude the proof
of Theorem 1.1 from this.

###### Proof of Theorem 1.1.

Let $\mu$ and $\nu$ be as in the hypothesis, with $\dim\mu>1>\dim\nu$. For a
contradiction, suppose that

$\\{|\lambda_{1}(A_{i})|:\ i\in\Gamma\\}\cup\\{|\lambda_{2}(B_{j})|:\
j\in\Lambda\\}$

is not an arithmetic set. It is not difficult to see that there must then
exist a pair $(i,j)\in\Gamma\times\Lambda$ such that
$\frac{\log|\lambda_{1}(A_{i})|}{\log|\lambda_{2}(B_{j})|}\not\in\mathbb{Q}$.

Let $\varepsilon>0$ and let $N\in\mathbb{N}$ be large enough as in Claims 4.3
and 4.4 and Theorem 3.2. Apply the affine change of coordinates offered by
Proposition 4.1. First of all, by [19, Lemma 4.3], we have

$\dim\mu=\liminf_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{N}H_{N}(\mu^{\mathtt{i},k})$

for $\bar{\mu}$-almost every $\mathtt{i}\in\Gamma^{\mathbb{N}}$, so by (4.3)
and the previous claims we have

$\displaystyle\liminf_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{N}H_{N}(\mu^{\mathtt{i},k}*\nu_{\mathtt{j}|_{i_{k}}})$
$\displaystyle\geq\ $
$\displaystyle\liminf_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\left(\frac{1}{N}H_{N}(\pi^{2}\mu^{\mathtt{i},k}*\pi^{2}\nu_{\mathtt{j}|_{i_{k}}})+\frac{1}{N}H_{N}(\mu^{\mathtt{i},k})-\frac{1}{N}H_{N}(\pi^{2}\mu^{\mathtt{i},k})-O(1/N)\right)$
$\displaystyle\geq\ $
$\displaystyle\min\\{1,\dim\mu-1+\dim\nu\\}+\dim\mu-(\dim\mu-1)-\varepsilon$
$\displaystyle=\ $ $\displaystyle\min\\{2,\dim\mu+\dim\nu\\}-\varepsilon$

for $\bar{\mu}$-almost every $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and
$\bar{\nu}$-almost every $\mathtt{j}\in\Lambda^{\mathbb{N}}$. It now follows
from Theorem 3.2 that
$\dim(\mu*\nu)\geq\min\\{2,\dim\mu+\dim\nu\\}-2\varepsilon$, which is a
contradiction if $\varepsilon$ is small enough. ∎

##  5\. The scenery of the self-affine measure

In this section, our aim is to prove Proposition 4.1, whence we assume
throughout the section that $\mu$ is a self-affine measure as in Theorem 1.1
with $\dim\mu>1$. The arguments of this section were inspired by the work of
Kempton [22] on a similar result for a more special class of self-affine
measures, and some of our lemmas are analogous to those in [22].

###  5.1. Restrictions of $\mu$ on thin rectangles

The strong separation condition asserts the existence of a bounded open set
$V\neq\emptyset$ whose closure is mapped into disjoint subsets of $V$ by the
maps $\varphi_{i}$. For the sake of notational simplicity, we suppose that
$V=B(0,1)$; since all the statements in the following are local in nature,
everything works also for a general $V$ by restricting onto small balls
centered in the point of interest.

Let $\mathtt{i}\in\Gamma^{\mathbb{N}}$, let $B$ be a small ball centered at
$\Pi(\mathtt{i})$ and let $n$ be the largest integer so that
$\varphi_{\mathtt{i}|_{n}}(B(0,1))\supseteq B$. By the strong separation
condition, we have
$\mu_{B}=\varphi_{\mathtt{i}|_{n}}\varphi_{\mathtt{i}|_{n}}^{-1}\mu_{B}=\varphi_{\mathtt{i}|_{n}}\mu_{\varphi_{\mathtt{i}|_{n}}^{-1}B}$.
Therefore, in order for us to understand the magnifications $\mu_{B}$, we have
to understand the measures $\mu_{\varphi_{\mathtt{i}|_{n}}^{-1}B}$, the
restrictions of $\mu$ on the thinner and thinner ellipses
$\varphi_{\mathtt{i}|_{n}}^{-1}B$ whose major axes have length comparable to
$1$. It is intuitive that these measures should approximate the slice measures
of $\mu$ in different directions, as $n\rightarrow\infty$. But since slices
can be defined as limits of the measures supported on thinner and thinner
tubes, it is better to work with rectangles instead of ellipses for a moment.

For $\mathtt{i}\in\Gamma^{\mathbb{N}}$, $\theta\in\mathbb{RP}^{1}$, $r_{2}\geq
r_{1}\geq 0$, write $Y_{\mathtt{i},\theta,r_{1},r_{2}}$ for the rectangle
centered at $\Pi(\mathtt{i})$ with sidelengths $2^{-r_{1}}\geq 2^{-r_{2}}$ and
the longer side oriented in the direction $\theta$. For a rectangle $Y$ with
center at $x$ and major side oriented in the direction $\theta$, write $H_{Y}$
for the map which translates $x$ to the origin and stretches $T_{x}Y$ onto
$R_{\theta}^{-1}[-1,1]^{2}$. It follows from Lemma 2.23 and standard geometric
measure theory that the measures
$H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}\mu_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}$
have a fiber structure in the following sense.

######  Lemma 5.1.

For $\bar{\mu}\times\mu_{F}$-a.e.
$(\mathtt{i},\theta)\in\Gamma^{\mathbb{N}}\times\mathbb{RP}^{1}$ and any
$\delta,\varepsilon,c,r_{1}>0$, there exists $t>0$ such that the following
holds:

If $(Q_{r_{2}})_{r_{2}\geq 0}$ is a sequence of squares that contain the
origin with one side parallel to $\theta$, $|Q_{r_{2}}|\geq c>0$ and

$H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}\mu_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}(Q_{r_{2}})\geq\delta$

for every $r_{2}\geq t$, then

$d_{\rm
LP}\left(\pi_{\theta}((H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}\mu_{Y_{\mathtt{i},\theta,r_{1},r_{2}}})_{Q_{r_{2}}}),\
(R_{\theta}^{-1}\mu_{\mathtt{i},\theta,r_{1}})_{\pi_{\theta}Q_{r_{2}}}\right)<\varepsilon$

for every $r_{2}\geq t$.

For the proof, we record the following elementary observation.

######  Lemma 5.2.

For any $r,\delta>0$, the following holds for all small enough
$\varepsilon\geq\varepsilon^{\prime}>0$:

If $\mu$ and $\nu$ are probability measures on $[-1,1]^{d}$ with $d_{\rm
LP}(\mu,\nu)<\varepsilon^{\prime}$ and $B=B(x,r)$ is a closed ball with
$\min\\{\mu(B),\nu(B)\\}\geq\delta$ and
$\nu(B(x,r+\varepsilon^{\prime}))\leq\nu(B(x,r-\varepsilon^{\prime}))+\varepsilon$,
then

$d_{\rm LP}(\mu_{B},\nu_{B})<O(\varepsilon/\delta).$

###### Proof of Lemma 5.1.

Since $\dim\mu>1$, it follows from Theorems 2.5 and 2.6 that
$\dim\mu_{\mathtt{i},\theta}>0$ for $\bar{\mu}\times\mu_{F}$-almost every
$(\mathtt{i},\theta)$. In particular, the slices $\mu_{\mathtt{i},\theta}$ are
non-atomic. Recall that the slices $\mu_{\mathtt{i},\theta}$ have been
translated so that the origin belongs to their supports. For
$\bar{\mu}\times\mu_{F}$-a.e. $(\mathtt{i},\theta)$, it follows from non-
atomicity of $\mu_{\mathtt{i},\theta}$ that

$\inf\\{\mu_{\mathtt{i},\theta}([a,b]):\ a\leq 0\leq b,\ b-a\geq c\\}>0.$

Thus, for a given $\varepsilon>0$, there exists $c^{\prime}>0$ and a set
$A_{\varepsilon}$ with $\bar{\mu}\times\mu_{F}(A_{\varepsilon})>1-\varepsilon$
such that $\mu_{\mathtt{i},\theta}([a,b])\geq c^{\prime}$ for every
$(\mathtt{i},\theta)\in A_{\varepsilon}$ and $a\leq 0\leq b$ with $b-a\geq c$.
Since $\bar{\mu}\times\mu_{F}(\bigcup_{n\in\mathbb{N}}A_{1/n})=1$, it suffices
to prove the statement for $(\mathtt{i},\theta)\in A_{\varepsilon}$, for a
given $\varepsilon>0$.

Now, let $E_{r_{2}}=\pi_{\theta^{\perp}}^{-1}(\pi_{\theta^{\perp}}Q_{r_{2}})$.
Lemma 2.23 translated to the language of this section states that

$d_{\rm
LP}\left(\pi_{\theta}(H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}\mu_{Y_{\mathtt{i},\theta,r_{1},r_{2}}})_{E_{r_{2}}},\
R_{\theta}^{-1}\mu_{\mathtt{i},\theta,r_{1}}\right)\rightarrow 0.$

See Figure 1. Since
$E_{r_{2}}\cap\pi_{\theta}^{-1}(\pi_{\theta}Q_{r_{2}})=Q_{r_{2}}$,
$\mu_{\mathtt{i},\theta}(\pi_{\theta}Q_{r_{2}})\geq c^{\prime}$ and
$\mu_{\mathtt{i},\theta}$ is non-atomic, Lemma 5.2 asserts that also

$d_{\rm
LP}\left(\pi_{\theta}(H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}\mu_{Y_{\mathtt{i},\theta,r_{1},r_{2}}})_{Q_{r_{2}}},\
(R_{\theta}^{-1}\mu_{\mathtt{i},\theta,r_{1}})_{\pi_{\theta}Q_{r_{2}}}\right)\rightarrow
0.$

∎

$\theta$$\longleftarrow_{\pi_{\theta}}$$\pi_{\theta}Q_{r_{2}}$$(0,0)$$Q_{r_{2}}$$H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}Y_{\mathtt{i},\theta,r_{1},r_{2}}$$E_{r_{2}}$
Figure 1. The measure
$H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}\mu_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}$
projects close to a slice of $\mu$ under $\pi_{\theta}$, even when restricted
onto $E_{r_{2}}$.

We will next relate the measures $\mu^{\mathtt{i},k}$ to the measures
$H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}\mu_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}$.
To make statements more economic, we introduce some additional notation.

###  5.2. Magnifications of $\mu$

For $\mathtt{a}\in\Gamma^{*}$, let
$A_{\mathtt{a}}=U_{\mathtt{a}}D_{\mathtt{a}}V_{\mathtt{a}}^{-1}$ denote the
singular value decomposition, where $U_{\mathtt{a}},V_{\mathtt{a}}$ are
orthogonal and $D_{\mathtt{a}}={\rm
diag}(\alpha_{1}(A_{\mathtt{a}}),\alpha_{2}(A_{\mathtt{a}}))$. Fix a large
integer $N$, and for $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and $k\in\mathbb{N}$,
let $\ell_{k}=\ell_{k}(\mathtt{i})$ be any increasing sequence such that
$\lim_{k\rightarrow\infty}\ell_{k}=\infty$ and

(5.1)
$B(\Pi(\mathtt{i}),2^{-kN})\cap\Pi(\Gamma^{\mathbb{N}})=B(\Pi(\mathtt{i}),2^{-kN})\cap\varphi_{\mathtt{i}|_{\ell_{k}}}(B(0,1))$

for every $k$. Such a sequence exists for every $\mathtt{i}$ by the strong
separation condition. Throughout the following, we will use the short-hand
notation

$Q_{\mathtt{i},\theta,k}:=Y_{\sigma^{\ell_{k}}\mathtt{i},\theta,kN+\log\alpha_{1}(\mathtt{i}|_{\ell_{k}}),kN+\log\alpha_{2}(\mathtt{i}|_{\ell_{k}})}.$

######  Proposition 5.3.

For every $(\mathtt{i},\theta)\in\Gamma^{\mathbb{N}}\times\mathbb{RP}^{1}$,
there exists an integer $m\geq 0$ and a sequence of non-singular linear maps
$(L_{\mathtt{i},\theta,k})_{k\in\mathbb{N}}$ such that

$S_{kN+m}T_{\Pi(\mathtt{i})}\mu=S_{m}U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}\mu_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}$

for all large enough $k$.

An important point of the proposition is that the direction $\theta$ can be
chosen arbitrarily. In proving this we require the following geometric lemma,
analogous to [22, Lemma 8.2]. Write $E_{\mathtt{i},\theta,k}\subseteq
Q_{\mathtt{i},\theta,k}$ for the largest ellipse contained in
$Q_{\mathtt{i},\theta,k}$.

######  Lemma 5.4 (Figure 2).

For $\mu\times\mu_{F}$-almost every $(\mathtt{i},\theta)$, there exists
$0<C<1$ such that

$CT_{\Pi(\sigma^{\ell_{k}}\mathtt{i})}E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}\subseteq
T_{\Pi(\sigma^{\ell_{k}}\mathtt{i})}Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}$

for all large enough $k$.

Figure 2. The ellipse
$E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}$ (light gray) fits
completely inside the rectangle
$Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}$ after the axes are
scaled by some constant $C$.

###### Proof.

For any $\theta$, $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and $n\in\mathbb{N}$, we
have

(5.2) $\tan(d(A_{\mathtt{i}|_{n}}^{-1}\theta,\
\theta(A_{\mathtt{i}|_{n}}^{-1})))=\frac{\alpha_{1}(\mathtt{i}|_{n})}{\alpha_{2}(\mathtt{i}|_{n})}\tan(d(\theta,\
\theta(A_{\mathtt{i}|_{n}})^{\perp})).$

See Figure 3.

$\ell$$1$$A_{\mathtt{i}|_{n}}^{-1}\
\mapsto$$\alpha_{2}(\mathtt{i}|_{n})^{-1}\ell$$\alpha_{1}(\mathtt{i}|_{n})^{-1}$$\beta$$\beta^{\prime}$
Figure 3. Here $\beta=d(\theta,\ \theta(A_{\mathtt{i}|_{n}})^{\perp})$ which
is taken to $\beta^{\prime}=d(A_{\mathtt{i}|_{n}}^{-1}\theta,\
\theta(A_{\mathtt{i}|_{n}}^{-1}))$ by $A_{\mathtt{i}|_{n}}^{-1}$. If the side
of length $1$ is parallel to $\theta(A_{\mathtt{i}|_{n}})^{\perp}$ (which is
the major expanding direction of $A_{\mathtt{i}|_{n}}^{-1}$), then its image
through $A_{\mathtt{i}|_{n}}^{-1}$ is of length
$\alpha_{1}(\mathtt{i}|_{n})^{-1}$.

Write $a=\alpha_{2}(\mathtt{i}|_{n})$ and $b=\alpha_{1}(\mathtt{i}|_{n})$. Let
$X=[-b,b]\times[-a,a]$ and $Y=R_{\eta}^{-1}([-Cb,Cb]\times[-Ca,Ca])$ with
$\eta=d(A_{\mathtt{i}|_{n}}^{-1}\theta,\ \theta(A_{\mathtt{i}|_{n}}^{-1}))$.
It is not difficult to see that $Y\subseteq X$ if we have $\sin\eta\cdot
Ca+Cb\leq b$, or equivalently,

$\sin\eta\leq\frac{(1-C)b}{Ca}.$

Since $\eta\rightarrow 0$ as $n\rightarrow\infty$, we have $\sin\eta\leq
2\tan\eta$ for large enough $n$. Therefore, we have

$\sin\eta\leq\frac{2b}{a}\tan(d(\theta,\
\theta(A_{\mathtt{i}|_{n}})^{\perp}))\leq\frac{(1-C)b}{Ca},$

when we choose $C\leq\frac{1}{4\tan(d(\theta,\
\theta(\mathtt{i})^{\perp}))+1}$. ∎

###### Proof of Proposition 5.3.

Let $(\mathtt{i},\theta)\in\Gamma^{\mathbb{N}}\times\mathbb{RP}^{1}$, let $k$
be large, and let $\ell_{k}$ be defined as in (5.1). Note that with the above
notation,
$(\varphi_{\mathtt{i}|_{\ell_{k}}})^{-1}B(\Pi(\mathtt{i}),2^{-(k-1)N})=E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}$.
Write
$B_{\mathtt{i}|_{\ell_{k}}}:=U_{\mathtt{i}|_{\ell_{k}}}D^{-1}_{\mathtt{i}|_{\ell_{k}}}U_{\mathtt{i}|_{\ell_{k}}}^{-1}$
and note that $S_{(k-1)N}B_{\mathtt{i}|_{\ell_{k}}}^{-1}$ is the linear map
which scales the origocentric ellipse
$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}T_{\varphi_{\mathtt{i}|_{\ell_{k}}}^{-1}\pi(\mathtt{i})}E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}$
onto $B(0,1)$ without rotating it.

By the strong separation condition, with this notation we may write

$\displaystyle S_{(k-1)N}T_{\Pi(\mathtt{i})}\mu$
$\displaystyle=S_{(k-1)N}T_{\pi(\mathtt{i})}\varphi_{\mathtt{i}|_{\ell_{k}}}\mu_{E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}$
$\displaystyle=S_{(k-1)N}B_{\mathtt{i}|_{\ell_{k}}}^{-1}B_{\mathtt{i}|_{\ell_{k}}}A_{\mathtt{i}|_{\ell_{k}}}T_{\varphi_{\mathtt{i}|_{\ell_{k}}}^{-1}\pi(\mathtt{i})}\mu_{E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}$
$\displaystyle=S_{(k-1)N}B_{\mathtt{i}|_{\ell_{k}}}^{-1}U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}T_{\varphi_{\mathtt{i}|_{\ell_{k}}}^{-1}\pi(\mathtt{i})}\mu_{E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}$
$\displaystyle=U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}H_{Q_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}\mu_{E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}.$

by switching the order of scaling and rotation in the last equality.

Let $0<C<1$ be the constant given by Lemma 5.4 so that

$CT_{\Pi(\sigma^{\ell_{k}}\mathtt{i})}E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}\subseteq
T_{\Pi(\sigma^{\ell_{k}}\mathtt{i})}Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}.$

Then $S_{-\log
C}H_{Q_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}\mu_{E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}=S_{-\log
C}H_{Q_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}\mu_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}$;
see Figure 4.

$Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}$$E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}$

$H_{\ldots}\,\rightarrow$

Figure 4. The dynamics of
$(A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta)_{k\in\mathbb{N}}$ are easier to
control than those of
$(\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}))_{k\in\mathbb{N}}$ even though the
sequences are topologically asymptotic, so we want to consider restrictions of
$\mu$ on $Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}$ instead of
$E_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}$.

Writing

$L_{\mathtt{i},\theta,k}:=H_{Q_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}^{-1},$

we have obtained the representation

$S_{(k-1)N-\log C}T_{\Pi(\mathtt{i})}\mu=S_{-\log
C}U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}\mu_{Q_{\mathtt{i},A^{-1}_{\mathtt{i}|_{\ell_{k}}},k}}.$

This proves the statement with $m:=\lceil-\log C\rceil$. ∎

In particular, since
$2^{kN}T_{\Pi(\mathtt{i})}D_{kN+m}(\Pi(\mathtt{i}))=T_{2^{kN}\Pi(\mathtt{i})\mod
1}D_{m}(2^{kN}\Pi(\mathtt{i})\mod 1)$, where we write

$(x,y)\mod 1:=(x\mod 1,y\mod 1)$

for $(x,y)\in\mathbb{R}^{2}$, and $D_{kN+m}(\Pi(\mathtt{i}))\subseteq
B(\Pi(\mathtt{i}),2^{-kN-m})$, we have

$\mu^{\mathtt{i},k+m}=(U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}\mu_{Q_{\mathtt{i},A^{-1}_{\mathtt{i}|_{\ell_{k}}},k}})^{T_{2^{kN}\Pi(\mathtt{i})\mod
1}D_{m}(2^{kN}\Pi(\mathtt{i})\mod 1)}$

for every large enough $k$.

###  5.3. The distortion
$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}$

If the map
$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}$
were just a rotation, then all that would be left to obtain the fiber
structure of $\mu^{\mathtt{i},k+m}$ were to apply Lemma 5.1. However, it turns
out that this is not necessarily the case, and
$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}$
instead takes squares onto parallelograms in a way which depends on
$\mathtt{i}$ and $\theta$.

To overcome the difficulties brought by this additional distortion, we begin
by breaking $U_{\mathtt{i}|_{n}}V_{\mathtt{i}|_{n}}^{-1}$ into three
components: The reflection, and two rotations which do not reflect.

For a linear map $A:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ and
$\theta\in\mathbb{RP}^{1}$, let $A|_{\theta}:\
\theta\rightarrow\mathbb{R}^{2}$ denote the map obtained by restricting $A$
onto $\theta$. Let $O\subset\Gamma^{\mathbb{N}}\times\mathbb{RP}^{1}$ be the
open set of those $(\mathtt{i},\theta)$ for which $\langle\pi^{2}(x)\ |\
\pi^{2}\circ A_{i_{0}}^{-1}(x)\rangle<0$ for some (equivalently every)
$x\in\theta$. Define the function $\rho:\
\mathbb{N}\times\Gamma\times\mathbb{RP}^{1}\rightarrow\\{-1,1\\}$,

(5.3)
$\rho(n,(\mathtt{i},\theta))=\prod_{k=1}^{n}(-1)^{\chi_{O}(M^{k}(\mathtt{i},\theta))}.$

This map captures the reflections done by $A_{\mathtt{i}|_{n}}^{-1}$ on the
line $\theta$, or by $A_{\mathtt{i}|_{n}}$ on the line
$A_{\mathtt{i}|n}^{-1}\theta$. Indeed, for any $x\in\theta$, we may decompose
$A_{\mathtt{i}|_{n}}^{-1}$ as

(5.4)
$A_{\mathtt{i}|_{n}}^{-1}x=\|A_{\mathtt{i}|_{n}}^{-1}x\|R_{A_{\mathtt{i}|_{n}}^{-1}\theta}^{-1}\rho(n,(\mathtt{i},\theta))R_{\theta}x.$

In other words, first rotate $x$ to the $y$-axis, apply the possible
reflections, rotate the $y$-axis onto the line
$A_{\mathtt{i}|_{n}}^{-1}\theta$, and finally scale. The map $\rho$ is easily
seen to satisfy the _cocycle equation_

$\rho(n+k,(\mathtt{i},\theta))=\rho(n,M^{k}(\mathtt{i},\theta))\rho(k,(\mathtt{i},\theta)).$

######  Lemma 5.5.

Let $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and let
$U_{\mathtt{i}|_{n}}D_{\mathtt{i}|_{n}}V^{-1}_{\mathtt{i}|_{n}}$ be the
singular value decomposition of $A_{\mathtt{i}|_{n}}$. For any
$\theta\in\mathbb{RP}^{1}\setminus\\{\theta(\mathtt{i})\\}$, we have

$\|U_{\mathtt{i}|_{n}}V_{\mathtt{i}|_{n}}^{-1}|_{A_{\mathtt{i}|_{n}}^{-1}\theta}-R_{\theta(A_{\mathtt{i}|_{n}})^{\perp}}^{-1}\rho(n,(\mathtt{i},\theta))R_{A_{\mathtt{i}|_{n}}^{-1}\theta}|_{A^{-1}_{\mathtt{i}|_{n}}\theta}\|\rightarrow
0$

as $n\rightarrow\infty$.

###### Proof.

Let
$B_{\mathtt{i}|_{n}}:=U_{\mathtt{i}|_{n}}D_{\mathtt{i}|_{n}}^{-1}U_{\mathtt{i}|_{n}}^{-1}$
and note that

$A_{\mathtt{i}|_{n}}=B_{\mathtt{i}|_{n}}^{-1}B_{\mathtt{i}|_{n}}A_{\mathtt{i}|_{n}}^{-1}=B_{\mathtt{i}|_{n}}^{-1}U_{\mathtt{i}|_{n}}V_{\mathtt{i}|_{n}}^{-1}.$

In particular,
$U_{\mathtt{i}|_{n}}V_{\mathtt{i}|_{n}}^{-1}=B_{\mathtt{i}|_{n}}A_{\mathtt{i}|_{n}}$.

Now, by (5.4),

$A_{\mathtt{i}|_{n}}|_{A^{-1}_{\mathtt{i}|_{n}}\theta}=\|A_{\mathtt{i}|_{n}}|_{A^{-1}_{\mathtt{i}|_{n}}\theta}\|R_{\theta}^{-1}\rho(n,(\mathtt{i},\theta))R_{A_{\mathtt{i}|_{n}}^{-1}\theta}|_{A^{-1}_{\mathtt{i}|_{n}}\theta}.$

On the other hand,

$\left\|B_{\mathtt{i}|_{n}}|_{\theta}-(-1)^{k_{n}}\|A_{\mathtt{i}|_{n}}^{-1}|_{\theta}\|R_{\theta(\mathtt{i})^{\perp}}^{-1}R_{\theta}|_{\theta}\right\|\rightarrow
0$

as $n\rightarrow\infty$, for some sequence $(k_{n})_{n}\in\mathbb{N}$, by
Lemma 2.4. Since $B_{\mathtt{i}|_{n}}$ is positive definite and
$\theta\neq\theta(\mathtt{i})$, the sequence $(-1)^{k_{n}}$ has to be
eventually constant. By absorbing the eventual value of this sequence to the
definition of $\rho(n,(\mathtt{i},\theta))$, we may without loss of generality
suppose that

$\lim_{n\rightarrow\infty}\|A_{\mathtt{i}|_{n}}^{-1}|_{\theta}\|^{-1}B_{\mathtt{i}|_{n}}|_{\theta}=R_{\theta(\mathtt{i})^{\perp}}^{-1}R_{\theta}|_{\theta}.$

Thus

$\|B_{\mathtt{i}|_{n}}A_{\mathtt{i}|_{n}}|_{A^{-1}_{\mathtt{i}|_{n}}\theta}-R_{\theta(\mathtt{i})^{\perp}}^{-1}\rho(n,(\mathtt{i},\theta))R_{A_{\mathtt{i}|_{n}}^{-1}\theta}|_{A^{-1}_{\mathtt{i}|_{n}}\theta}\|\rightarrow
0.$

∎

It remains to study the behaviour of linear map $L_{\mathtt{i},\theta,k}$ as
$k\rightarrow\infty$, in the statement of Proposition 5.3. The content of the
following lemma is that
$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}$
takes
$H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}=R_{(A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta)^{\perp}}[-1,1]^{2}$
onto a parallelogram of bounded eccentricity and one side in direction
$\theta(\mathtt{i})$.

######  Lemma 5.6.

For every $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and
$\theta\in\mathbb{RP}^{1}\setminus\\{\theta(\mathtt{i})\\}$,

$\|U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}-R_{\theta(\mathtt{i})^{\perp}}^{-1}\rho(\ell_{k},(\mathtt{i},\theta))F_{\theta,\mathtt{i}}^{k}R_{A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta}\|\rightarrow
0$

as $k\rightarrow\infty$, where
$F_{\theta,\mathtt{i}}^{k}=\begin{bmatrix}a_{k}&b_{k}\cdot\tan(d(\theta,\
\theta(\mathtt{i})^{\perp}))\\\ 0&1\end{bmatrix}$ for some
$a_{k},b_{k}\in\\{-1,1\\}$.

###### Proof.

Recall that
$L_{\mathtt{i},\theta,k}=H_{Q_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}^{-1}$.
As the map $L_{\mathtt{i},\theta,k}$ acts on the set
$R_{A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta}^{-1}[-1,1]^{2}$, it is first taken
to the thin rectangle
$H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}^{-1}R_{A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta}^{-1}[-1,1]^{2}$
(see Figure 5), and then stretched onto a parallelogram by
$H_{Q_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}$.

$H_{Q_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}\,\rightarrow$

Figure 5. The set
$H_{Q_{\mathtt{i},\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),k}}^{-1}R_{\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1})^{\perp}}^{-1}[-1,1]^{2}$
is depicted as the thin white rectangle, while the set
$H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}^{-1}R_{(A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta)^{\perp}}^{-1}[-1,1]^{2}$
is depicted as the thin gray rectangle.

The angle between these thin rectangles is
$d(\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),\
A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta)$, so by (5.2) and basic geometry, this
parallelogram tends to
$R_{\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1})}^{-1}F_{\theta,\mathtt{i}}^{k}[-1,1]^{2}$
as $k\rightarrow\infty$, where in the definition of
$F^{k}_{\theta,\mathtt{i}}$ we have $a_{k}=1$ and $b_{k}\in\\{-1,1\\}$ depends
on the order of $\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1})$ and
$A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta$. In particular,

$\|L_{\mathtt{i},\theta,k}-R_{\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1})}^{-1}F_{\theta,\mathtt{i}}^{k}R_{A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta}\|\rightarrow
0$

as $k\rightarrow\infty$. Using the fact that
$d(\theta(A_{\mathtt{i}|_{\ell_{k}}}^{-1}),\
A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta)\rightarrow 0$, Lemma 5.5 and
incorporating the possible reflection for the line $\theta(\mathtt{i})$ as the
value of $a_{k}$ in the definition of $F_{\theta,\mathtt{i}}^{k}$ completes
the proof. ∎

###  5.4. Fiber structure for $\mu^{\mathtt{i},k}$

We will now explain how to combine Proposition 5.3 and Lemmas 5.1 and 5.6 to
prove Proposition 4.1 and to obtain the fiber structure of
$\mu^{\mathtt{i},k}$. Recall that the problem is the additional distortion
brought by
$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}$;
see Figure 6.

$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}$$T_{2^{kN}\Pi(\mathtt{i})\mod
1}D_{m}(2^{kN}\Pi(\mathtt{i})\mod 1)$$(0,0)$ Figure 6. For us to be able to
directly apply Lemma 5.1, the above parallelogram would have to be the unit
square.

However, because of our freedom in choosing the direction $\theta$, this issue
can be resolved in the following way. Essentially, we perform a change of
coordinates so that $\mu_{F}$ gives positive mass to a small neighbourhood of
the $y$-axis, and all of the directions $\theta(\mathtt{i})$ are pulled very
close to the $x$-axis. As a result of this change, the parallelogram in Figure
6 is taken within Hausdorff distance $\varepsilon$ of the unit square.

Let $\varepsilon_{1}>0$ be a small number. We will later see how small it has
to be. Let $A$ be a linear map such that $0^{\perp}\in A({\rm spt}\,\mu_{F})$
and $d(A\theta(\mathtt{i}),0)<\varepsilon_{1}$ for all
$\mathtt{i}\in\Gamma^{\mathbb{N}}$. Choosing such an $A$ is possible since as
was seen in the proof of Lemma 2.3, the domination condition ensures that
${\rm
spt}\,\mu_{F}=\theta^{-}(\Gamma^{\mathbb{N}})\neq\theta(\Gamma^{\mathbb{N}})$.
This map will be the change of coordinates in the statement of Proposition
4.1.

Upon changing coordinates by $A$ we replace the IFS $\Phi$ by the conjugate
IFS $\\{A\varphi_{i}A^{-1}\\}_{i\in\Gamma}$, $\mu$ by the self-affine measure
$A\mu$, and the Furstenberg measure $\mu_{F}$ by the measure $A\mu_{F}$ (which
is the Furstenberg measure induced by $A\mu$).

$\varphi_{\mathtt{a}}(B(0,1))$$\theta_{0}=0$$\theta\in{\rm spt}\,\mu_{F}$

$A\varphi_{\mathtt{a}}(B(0,1))$$0$$A\theta$

Figure 7. If the ellipse $\varphi_{\mathtt{a}}(B(0,1))$ is thin enough w.r.t.
$\operatorname{diam}{\rm spt}\,\mu_{F}$, it remains thin after mapping it
through $A$ even though many directions $\theta\in{\rm spt}\,\mu_{F}$ are
pulled very close to $\theta_{0}^{\perp}$.

Afterwards, omitting the map $A$ from the notation, we have

(5.5) $\displaystyle d({\rm spt}\,\mu_{F},0^{\perp})$
$\displaystyle<\varepsilon_{1}\ \text{and}$ (5.6) $\displaystyle
d(\theta(\mathtt{i}),0)$ $\displaystyle<\varepsilon_{1}\ \text{for every}\
\mathtt{i}\in\Gamma^{\mathbb{N}}.$

Let $F_{\theta,\mathtt{i}}^{k}$ be as in Lemma 5.6. It follows from (5.5) and
(5.6) that for every $\mathtt{i}\in\Gamma^{\mathbb{N}}$ and for $\theta$ in a
set of positive $\mu_{F}$-measure, the map $F_{\theta,\mathtt{i}}^{k}$ is
within distance $\varepsilon_{1}$ of $\begin{bmatrix}a_{k}&0\\\
0&1\end{bmatrix}$, and thus the map $\pi^{2}\circ
U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}$
is within distance $\varepsilon_{1}$ of
$\pi^{2}\circ\rho(\ell_{k},(\mathtt{i},\theta))R_{A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta}$
for large $k$, by Lemma 5.6; see Figure 8.

$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}$$(0,0)$$T_{2^{kN}\Pi(\mathtt{i})\mod
1}D_{m}(2^{kN}\Pi(\mathtt{i})\mod 1)$ Figure 8. After our change of
coordinates, the parallelogram
$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}$
is close to the unit square.

We are now ready to prove Proposition 4.1.

###### Proof of Proposition 4.1.

Let $\varepsilon>0$, let $\varepsilon_{1}>0$ be small with respect to
$\varepsilon$ and and apply the change of coordinates $A$ that was described
above.

Let $m$ be the constant given by Proposition 5.3. Our first goal is to ensure
that the origin is often close to the center of the square
$T_{2^{kN}\Pi(\mathtt{i})\mod 1}D_{m}(2^{kN}\Pi(\mathtt{i})\mod 1)$, so that
the square has relatively large mass which is required to apply Lemma 5.1. In
this proof, write

$\mathtt{i}^{k}:=2^{k}\Pi(\mathtt{i})\mod 1$

and note that

$S_{kN}T_{\Pi(\mathtt{i})}D_{kN+m}(\Pi(\mathtt{i}))=T_{\mathtt{i}^{kN}}D_{m}(\mathtt{i}^{kN}).$

For any $\delta>0$

$B(\Pi(\mathtt{i}),2^{kN+m}\delta)\subseteq D_{kN+m}(\Pi(\mathtt{i}))\
\text{if and only if}\ B(\mathtt{i}^{kN+m},\delta)\subseteq[-1,1]^{2}.$

We begin by showing that this inclusion holds for most $k$, for small enough
$\delta$.

By Fubini’s theorem, by applying a random translation to $\mu$ (which does not
affect the dimension of $\mu*\nu$), we may suppose that for $\bar{\mu}$-almost
every $\mathtt{i}$, the sequence $(\mathtt{i}^{kN+m})_{k\in\mathbb{N}}$
equidistributes for the Lebesgue measure on $[-1,1]^{2}$. In particular, for a
small enough $\delta_{0}>0$ there exists
$\mathcal{N}_{\varepsilon}^{1}\subseteq\mathbb{N}$ with
$\liminf_{n\rightarrow\infty}\frac{\\#(\mathcal{N}^{1}_{\varepsilon}\cap[0,n])}{n}\geq
1-\varepsilon/4$ such that

(5.7) $B(\mathtt{i}^{kN+m},\delta_{0})\subseteq[-1,1]^{2}$

for every $k\in\mathcal{N}_{\varepsilon}^{1}$. On the other hand, by [15,
Proposition 1.19],

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\log
S_{kN}T_{\Pi(\mathtt{i})}\mu(B(0,\delta_{0}))}{\log\delta_{0}}\leq\overline{\dim}_{\rm
loc}\mu(\Pi(\mathtt{i}))<\infty$

for $\bar{\mu}$-a.e. $\mathtt{i}$, so there exists a $\delta>0$ and a set
$\mathcal{N}_{\varepsilon}^{2}\subseteq\mathbb{N}$ such that
$\liminf_{n\rightarrow\infty}\frac{\\#(\mathcal{N}_{\varepsilon}^{2}\cap[0,n])}{n}\geq
1-\varepsilon/4$ and

(5.8) $S_{kN}T_{\Pi(\mathtt{i})}\mu(B(0,\delta_{0}))\geq\delta>0$

for every $k\in\mathcal{N}_{\varepsilon}^{2}$.

Now, for $k\in\mathcal{N}_{\varepsilon}^{1}\cap\mathcal{N}_{\varepsilon}^{2}$,
we have

$S_{kN}T_{\Pi(\mathtt{i})}\mu(T_{\mathtt{i}^{kN}}D_{m}(\mathtt{i}^{kN}))\geq\delta.$

The remaining task is now to ensure that the measure
$U_{\mathtt{i}|_{\ell_{k}}}V_{\mathtt{i}|_{\ell_{k}}}^{-1}L_{\mathtt{i},\theta,k}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}\mu_{Q_{\mathtt{i},A^{-1}_{\mathtt{i}|_{\ell_{k}}}\theta,k}}$
does not give too large mass around the boundaries of squares, which was
another requirement of Lemma 5.1.

For $s,t>0$ define the set

$A_{s,t}=\\{\nu\in\mathcal{P}(\mathbb{R}^{2}):\ \text{there exists a line}\
\ell\ \text{with}\ d(\ell,0)>\delta_{0}\ \text{and}\ \nu(\ell^{s})\geq t\\}.$

By Lemma 2.22, for $\bar{\mu}$-almost every $\mathtt{i}$, there exists
$\varepsilon^{\prime}>0$ such that

$\limsup_{n\rightarrow\infty}\frac{\\#\\{0\leq k\leq n:\
S_{kN}T_{\Pi(\mathtt{i})}\mu\in
A_{\varepsilon^{\prime},\varepsilon_{1}}\\}}{n}\leq\varepsilon/4.$

Indeed, if this was not the case, we could find a sequence of $N$-tangent
distributions $(P_{n})_{n\in\mathbb{N}}$ such that
$P_{n}(A_{1/n,\varepsilon_{1}})\geq\varepsilon/4$ by the fact that
$A_{1/n,\varepsilon_{1}}$ is closed. In particular, any accumulation point $P$
of this sequence (which is again an $N$-tangent distribution) would have

$P\left(\bigcap_{n}A_{1/n,\varepsilon_{1}}\right)=\lim_{n\rightarrow\infty}P(A_{1/n,\varepsilon_{1}})\geq\lim_{n\rightarrow\infty}\limsup_{m\rightarrow\infty}P_{m}(A_{1/n,\varepsilon_{1}})\geq\varepsilon/4,$

contradicting Lemma 2.22. Let $\mathcal{N}_{\varepsilon}^{3}$ be such that for
each $k\in\mathcal{N}_{\varepsilon}^{3}$, $S_{kN}T_{\Pi(\mathtt{i})}\mu\in
A_{\varepsilon^{\prime},\varepsilon_{1}}$. Then
$\liminf_{n\rightarrow\infty}\frac{\\#(\mathcal{N}_{\varepsilon}^{3}\cap[0,n])}{n}\geq
1-\varepsilon/4$.

Let $X$ be the set given by Lemma 5.1 (applied with $c=\delta_{0}/2$), such
that $\bar{\mu}\times\mu_{F}(X)>1-\varepsilon_{1}$ and for all large enough
$r_{2}$ and every $(\mathtt{i},\theta)\in X$, we have

$d_{\rm
LP}\left(\pi_{\theta}((H_{Y_{\mathtt{i},\theta,r_{1},r_{2}}}\mu_{Y_{\mathtt{i},\theta,r_{1},r_{2}}})_{Q_{r_{2}}}),\
(R_{\theta}^{-1}\mu_{\mathtt{i},\theta,r_{1}})_{\pi_{\theta}Q_{r_{2}}}\right)<\varepsilon/2.$

Finally, for every $(\mathtt{i},\theta)$ we may let
$\mathcal{N}_{\varepsilon}^{4}\subseteq\mathbb{N}$ be the set of those $k$ for
which $M^{\ell_{k}}(\mathtt{i},\theta)\in X$. For
$\bar{\mu}\times\mu_{F}$-almost every $(\mathtt{i},\theta)$, we have
$\liminf_{n\rightarrow\infty}\frac{\\#\\{0\leq k\leq n:\
M^{k}(\mathtt{i},\theta)\in X\\}}{n}\geq 1-\varepsilon_{1}$ by Birkhoff’s
ergodic theorem, and since it is not difficult to verify from the definition
of $\ell_{k}$ (in (5.1)) that $\ell_{k}\leq O_{N}(k)$, we have
$\liminf_{n\rightarrow\infty}\frac{\\#(\mathcal{N}_{\varepsilon}^{4}\cap[0,n])}{n}\geq
1-\varepsilon/4$ if $\varepsilon_{1}$ is small enough.

Let now $\mathtt{i}\in\Gamma^{\mathbb{N}}$ be chosen from the set of full
$\bar{\mu}$-measure and $\theta\in\mathbb{RP}^{1}$ from a set of positive
$\mu_{F}$-measure such that for $(\mathtt{i},\theta)$, all of the sets
$\mathcal{N}_{\varepsilon}^{j}$ as above exist, and let
$\mathcal{N}_{\varepsilon}=\mathcal{N}_{\varepsilon}^{1}\cap\mathcal{N}_{\varepsilon}^{2}\cap\mathcal{N}_{\varepsilon}^{3}\cap\mathcal{N}_{\varepsilon}^{4}$.
By Proposition 5.3 and Lemma 5.6,

(5.9) $d_{\rm LP}(S_{kN+m}T_{\Pi(\mathtt{i})}\mu,\
S_{m}R_{\theta(\mathtt{i})^{\perp}}^{-1}\rho(\ell_{k},(\mathtt{i},\theta))F_{\theta,\mathtt{i}}^{k}R_{A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta}H_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}}\mu_{Q_{\mathtt{i},A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta,k}})<\varepsilon^{\prime}$

for every large enough $k$. Also, for every $k\in\mathcal{N}_{\varepsilon}$ we
have

$S_{kN}T_{\Pi(\mathtt{i})}\mu(T_{\mathtt{i}^{kN}}D_{m}(\mathtt{i}^{kN}))\geq\delta$

by (5.7) and (5.8), and

$S_{kN}T_{\Pi(\mathtt{i})}\mu((T_{\mathtt{i}^{kN}}D_{m}(\mathtt{i}^{kN}))^{\varepsilon^{\prime}})\leq
S_{kN}T_{\Pi(\mathtt{i})}\mu(T_{\mathtt{i}^{kN}}D_{m}(\mathtt{i}^{kN}))+\varepsilon_{1}$

by the definition of $\mathcal{N}_{\varepsilon}^{3}$. Since
$d(\theta(\mathtt{i})^{\perp},0^{\perp})<\varepsilon_{1}$ and

$\left\|F_{\mathtt{i},\theta}^{k}-\begin{bmatrix}a_{k}&0\\\
0&1\end{bmatrix}\right\|<\varepsilon_{1}$

by the properties (5.5) and (5.6) of our change of coordinates, we have by
Lemma 5.2, by the definition of $\mathcal{N}_{\varepsilon}^{4}$ and (5.9) that

$d_{\rm LP}\left(\pi^{2}\mu^{\mathtt{i},k+m},\
\left(\rho(\ell_{k},(\mathtt{i},\theta))\mu_{M^{\ell_{k}}(\mathtt{i},\theta),kN+\log\alpha_{1}(\mathtt{i}|_{\ell_{k}})}\right)^{I_{k}}\right)<\varepsilon/2+O(\varepsilon_{1}/\delta)<\varepsilon,$

for every $k\in\mathcal{N}_{\varepsilon}$, when
$I_{k}:=\pi^{2}T_{\mathtt{i}^{kN}}D_{m}(\mathtt{i}^{kN})$. ∎

##  6\. Bounding the local entropy averages

In this section, we prove Claims 4.2, 4.3 and 4.4. For this, we need to
understand the dynamics of the sequences
$(\pi^{2}\mu^{\mathtt{i},k})_{k\in\mathbb{N}}$ and
$(\pi^{2}\nu_{\mathtt{j}|_{i_{k}}})_{k\in\mathbb{N}}$.

###  6.1. Auxiliary suspension flows

We require a cocycle acting on $\Lambda^{\mathbb{N}}\times\mathbb{RP}^{1}$
similar to $\rho$ of the previous section, which we also denote by $\rho$
since it plays an identical role: Let
$O\subseteq\Lambda^{\mathbb{N}}\times\mathbb{RP}^{1}$ denote the open set of
those $(\mathtt{j},\theta)$ for which $\langle\pi_{\theta}\circ
B_{j_{0}}|\pi_{B_{j_{0}}^{*}\theta}\rangle<0$, and set

(6.1)
$\rho(k,(\mathtt{j},\theta))=\prod_{\ell=1}^{k}(-1)^{\chi[O](M_{*}^{\ell}(\mathtt{j},\theta))}.$

Define the Hölder continuous functions

$\displaystyle
f:\Gamma^{\mathbb{Z}}\times\mathbb{RP}^{1}\rightarrow\mathbb{R},$
$\displaystyle\
(\mathtt{i},\theta)\mapsto-\log\|A_{i_{0}}|_{A_{i_{0}}^{-1}\theta}\|$
$\displaystyle
g:\Lambda^{\mathbb{Z}}\times\mathbb{RP}^{1}\rightarrow\mathbb{R},$
$\displaystyle\ (\mathtt{j},\theta)=-\log\|B_{j_{0}}^{*}|_{\theta}\|$

and the sets

$\displaystyle Z_{\Phi}$ $\displaystyle:=\\{(\mathtt{i},\theta,u,t):\
\mathtt{i}\in\Gamma^{\mathbb{Z}},\theta\in\mathbb{RP}^{1},u\in\\{-1,1\\},0\leq
t\leq f(\mathtt{i},\theta)\\},$ $\displaystyle Z_{\Psi}$
$\displaystyle:=\\{(\mathtt{j},\theta,u,t):\
\mathtt{j}\in\Lambda^{\mathbb{Z}},\theta\in\mathbb{RP}^{1},u\in\\{-1,1\\},0\leq
t\leq g(\mathtt{j},\theta)\\},$

both equipped with the identifications
$(\mathtt{i},\theta,u,f(\mathtt{i},\theta))=(M(\mathtt{i},\theta),\rho(1,(\mathtt{i},\theta))u,0)$
and
$(\mathtt{j},\theta,u,g(\mathtt{j},\theta))=(M_{*}(\mathtt{j},\theta),\rho(1,(\mathtt{j},\theta))u,0)$.

Let $\mathcal{T}_{s}$ denote the flow induced by the positive reals on both
$Z_{\Phi}$ and $Z_{\Psi}$, given by

$\mathcal{T}_{s}:(\mathtt{k},\theta,u,t)\mapsto(\mathtt{k},\theta,u,t+s)$

for every $s\geq 0$ and $(\mathtt{k},\theta,u,t)\in Z_{\Phi}\cup Z_{\Psi}$.

Let $\gamma=\delta_{-1}/2+\delta_{1}/2$. Denote by $\lambda_{\Phi}$ the
measure $\bar{\mu}\times\mu_{F}\times\gamma\times\mathcal{L}$ restricted and
normalized on $Z_{\Phi}$, and let $\lambda_{\Psi}$ denote the normalized
restriction of $\bar{\nu}\times\nu_{F}^{*}\times\gamma\times\mathcal{L}$ on
$Z_{\Psi}$.

Denote by $\mathcal{Z}_{\Phi}$ and $\mathcal{Z}_{\Psi}$ the suspensions
$(Z_{\Phi},\lambda_{\Phi},\mathcal{T}_{s})$ and
$(Z_{\Psi},\lambda_{\Psi},\mathcal{T}_{s})$. It is not difficult to see that
they are measure-preserving. Since the skew-product maps

$\displaystyle(\mathtt{i},\theta,u)$
$\displaystyle\mapsto(M(\mathtt{i},\theta),\rho(1,(\mathtt{i},\theta))u),$
$\displaystyle(\mathtt{j},\theta,u)$
$\displaystyle\mapsto(M_{*}(\mathtt{j},\theta),\rho(1,(\mathtt{j},\theta))u)$

are $\bar{\mu}\times\mu_{F}\times\gamma$\- and
$\bar{\nu}\times\nu_{F}^{*}\times\gamma$-ergodic, respectively, by [28], it is
standard that these suspensions are also ergodic.

Let $\mathcal{Z}_{\Phi}^{\prime}=\pi^{1,2,4}\mathcal{Z}_{\Phi}$ and
$\mathcal{Z}_{\Psi}^{\prime}=\pi^{1,2,4}\mathcal{Z}_{\Psi}$. Here
$\pi^{1,2,4}$ denotes the projection onto the first, second and fourth
coordinate. The systems $\mathcal{Z}_{\Phi}^{\prime}$ and
$\mathcal{Z}_{\Psi}^{\prime}$ are suspension flows of
$\Gamma^{\mathbb{N}}\times\mathbb{RP}^{1}$ and
$\Lambda^{\mathbb{N}}\times\mathbb{RP}^{1}$ over $f$ and $g$, respectively,
and preserve the measures
$\lambda_{\Phi}^{\prime}:=\bar{\mu}\times\mu_{F}\times\mathcal{L}$ and
$\lambda_{\Psi}^{\prime}:=\bar{\nu}\times\nu_{F}^{*}\times\mathcal{L}$,
restricted and normalized on the sets $Z_{\Phi}^{\prime}=\pi^{1,2,4}Z_{\Phi}$
and $Z_{\Psi}^{\prime}=\pi^{1,2,4}Z_{\Psi}$.

The key observation we require on the dynamics of
$\mathcal{Z}_{\Phi}^{\prime}\times\mathcal{Z}_{\Psi}^{\prime}$ is the
following.

######  Proposition 6.1.

If the IFSs $\Phi$ and $\Psi$ are dominated and there exists a pair
$(i,j)\in\Gamma\times\Lambda$ with

$\frac{\log|\lambda_{1}(A_{i})|}{\log|\lambda_{2}(B_{j})|}\not\in\mathbb{Q},$

then the flow $\mathcal{Z}_{\Phi}^{\prime}\times\mathcal{Z}_{\Psi}^{\prime}$
is ergodic under the discrete-time map $\mathcal{T}_{N}$ for some
$N\in\mathbb{N}$.

Let $f^{\prime}:\Gamma^{\mathbb{Z}}\rightarrow(0,+\infty)$ denote the Hölder
continuous map $\mathtt{i}\mapsto
f(\mathtt{i}^{+},\theta^{-}(\mathtt{i}^{-}))$, and let

$Z_{\Phi}^{\prime\prime}=\\{(\mathtt{i},t):\
\mathtt{i}\in\Gamma^{\mathbb{Z}},0\leq t\leq f^{\prime}(\mathtt{i})\\}.$

Let $\lambda^{\prime\prime}_{\Phi}$ denote the measure
$\bar{\mu}\times\mathcal{L}$ restricted and normalized on
$Z_{\Phi}^{\prime\prime}$, and write
$\mathcal{Z}^{\prime\prime}_{\Phi}=(Z_{\Phi}^{\prime\prime},\mathcal{T}_{s},\lambda^{\prime\prime}_{\Phi})$
for the suspension of $\Gamma^{\mathbb{Z}}$ over $f^{\prime}$. Similarly, let
$\mathcal{Z}_{\Psi}^{\prime\prime}$ be the suspension of
$\Lambda^{\mathbb{Z}}$ over
$g^{\prime}(\mathtt{j}):=g(\mathtt{j}^{+},\theta^{*}(\mathtt{j}^{-}))$.

We will first show that the semiflow
$\mathcal{Z}_{\Phi}^{\prime\prime}\times\mathcal{Z}_{\Psi}^{\prime\prime}$ is
ergodic, and then use this to prove the ergodicity of
$\mathcal{Z}_{\Phi}^{\prime}\times\mathcal{Z}_{\Psi}^{\prime}$. Borrowing an
idea of Bowen [7], let $r:\Gamma^{\mathbb{Z}}\rightarrow\Gamma^{\mathbb{Z}}$
denote the function which replaces all the negative coordinates of $\ldots
i_{-1};i_{0}i_{1}\ldots$ by $i_{0}$. Then $f^{\prime}$ is cohomologous to
$h:=f^{\prime}\circ r$, that is,

(6.2) $f^{\prime}=h+u-u\circ\sigma,$

where $u$ is the continuous function defined by
$u(\mathtt{i})=\sum_{k=0}^{\infty}(f^{\prime}(\sigma^{k}\mathtt{i})-f^{\prime}(\sigma^{k}r(\mathtt{i})))$.
The sum converges since $f^{\prime}$ is Hölder. In particular, the flow
$\mathcal{Z}_{\Phi}^{\prime\prime}$ is conjugate to the suspension of
$\Gamma^{\mathbb{Z}}$ over $h$, denoted by
$\mathcal{Z}_{\Phi}^{h}=(Z_{\Phi}^{h},\mathcal{T}_{s},\lambda_{\Phi}^{h})$,
and the advantage of this is that the function $h$ depends only on the
positive coordinates of $\Gamma^{\mathbb{Z}}$.

######  Lemma 6.2.

The eigenvalues of the flow $\mathcal{Z}_{\Phi}^{\prime\prime}$ are contained
in the set

$\bigcap_{i\in\Gamma}(\log|\lambda_{1}(A_{i})|)^{-1}\mathbb{Q}.$

###### Proof.

Since $\mathcal{Z}_{\Phi}^{\prime\prime}$ is conjugate to
$\mathcal{Z}_{\Phi}^{h}$ and conjugate flows have the same eigenvalues, it
suffices to prove the statement for $\mathcal{Z}_{\Phi}^{h}$. Moreover, by
Lemma 2.11, we may replace $\mathcal{Z}_{\Phi}^{h}$ by
$(\mathcal{Z}_{\Phi}^{h})^{+}$ which we will in this proof continue to denote
by $\mathcal{Z}_{\Phi}^{h}$, for simplicity of notation.

Let $i\in\Gamma$, and let $\mathtt{i}_{0}=\ldots
iii\ldots\in\Gamma^{\mathbb{Z}}$.

Now,
$|\lambda_{1}(A_{i})|^{-1}=|\lambda_{2}(A_{i}^{-1})|=\|A_{i}^{-1}|_{\theta(\mathtt{i}_{0}^{-})}\|=\|A_{i}|_{\theta^{-}(\mathtt{i}_{0}^{-})}\|^{-1}$.
In particular,

$f^{\prime}(\mathtt{i}_{0})=-\log\|A_{i}|_{A_{i}^{-1}\theta^{-}(\mathtt{i}_{0}^{-})}\|=-\log|\lambda_{1}(A_{i})|.$

From (6.2) it is clear that also
$h(\mathtt{i}_{0})=-\log|\lambda_{1}(A_{i})|$.

Let $\beta\neq 0$ be an eigenvalue of $\mathcal{Z}_{\Phi}^{h}$. By Proposition
2.10, there exists a _continuous_ eigenfunction $\phi$ for $\beta$ on
$\mathcal{Z}_{\Phi}^{h}$, so we have

(6.3) $\phi(\mathcal{T}_{s}(\mathtt{i},t))=e(\beta s)\phi(\mathtt{i},t)$

for _every_ $(\mathtt{i},t)\in Z_{\Phi}^{h}$ and $s\geq 0$.

Since $\phi\neq 0$, we may let $\psi:Z_{\Phi}^{h}\rightarrow\mathbb{R}$ be the
real-valued function defined by $\phi(\mathtt{i},t)=e(\psi(\mathtt{i},t))$ for
every $(\mathtt{i},t)\in Z_{\Phi}^{h}$. We obtain from (6.3) that

$\psi(\mathcal{T}_{s}(\mathtt{i},t))=\beta
s+\psi(\mathtt{i},t)+n(\mathtt{i},t)$

for some integer-valued function $n:Z_{\Phi}^{h}\rightarrow\mathbb{Z}$.
Inserting the value $s=h(\mathtt{i})$, we obtain

$\psi(\sigma\mathtt{i},t)=\beta
h(\mathtt{i})+\psi(\mathtt{i},t)+n(\mathtt{i},t)$

which is equivalent to

$h(\mathtt{i})=\beta^{-1}n(\mathtt{i},t)+\beta^{-1}\psi(\mathtt{i},t)-\beta^{-1}\psi(\sigma\mathtt{i},t).$

In particular,
$h(\mathtt{i}_{0})=\beta^{-1}n(\mathtt{i}_{0},t)\in\beta^{-1}\mathbb{Z}$ since
$\mathtt{i}_{0}$ is a fixed point for $\sigma$. However, we saw above that
$h(\mathtt{i}_{0})=-\log|\lambda_{1}(A_{i})|$, whence it follows that
$\beta\in(\log|\lambda_{1}(A_{i})|)^{-1}\mathbb{Q}$. ∎

######  Lemma 6.3.

The eigenvalues of $\mathcal{Z}_{\Psi}^{\prime\prime}$ are contained in the
set

$\bigcap_{j\in\Lambda}(\log|\lambda_{2}(B_{j})|)^{-1}\mathbb{Q}$

###### Proof.

Let $\mathtt{j}_{0}=\ldots jjj\ldots$. Then $\mathtt{j}_{0}$ is a fixed point
for $\sigma$, and

$g^{\prime}(\mathtt{j}_{0})=-\log\|B_{j}^{*}|_{\theta^{*}(\mathtt{j}_{0}^{-})}\|=-\log|\lambda_{2}(B_{j}^{*})|=-\log|\lambda_{2}(B_{j})|$

and by replacing $\mathcal{Z}_{\Psi}^{\prime\prime}$ with a conjugate flow, we
can proceed exactly as in the proof of the previous claim. ∎

###### Proof of Proposition 6.1.

By Lemmas 6.2 and 6.3 and the assumption of Proposition 6.1, the flows
$\mathcal{Z}_{\Phi}^{\prime\prime}$ and $\mathcal{Z}_{\Psi}^{\prime\prime}$
have no common eigenvalues. Thus by Proposition 2.9, the product
$\mathcal{Z}_{\Phi}^{\prime\prime}\times\mathcal{Z}_{\Psi}^{\prime\prime}$ is
ergodic. Since it has at most countably many eigenvalues, there exists a real
number $c>0$ such that it is also ergodic under the discrete-time map
$\mathcal{T}_{c}$. By a change of coordinates, we may suppose further that
$c=N$ is an integer.

Now, since we have the domination assumption in place, the system
$\mathcal{Z}_{\Phi}^{\prime}\times\mathcal{Z}_{\Psi}^{\prime}$ is a factor of
the system
$\mathcal{Z}_{\Phi}^{\prime\prime}\times\mathcal{Z}_{\Psi}^{\prime\prime}$
through the map

$(\mathtt{i},t,\mathtt{j},s)\mapsto(\mathtt{i}^{+},\theta^{-}(\mathtt{i}^{-}),t,\mathtt{j}^{+},\theta^{*}(\mathtt{j}^{-}),s).$

Since factor maps preserve ergodicity, this proves the statement. ∎

###  6.2. Dynamics of the magnifications of $\mu$

We claim that the flows $\mathcal{Z}_{\Phi}$ and $\mathcal{Z}_{\Psi}$ capture
the dynamics of the sequences $(\pi^{2}\mu^{\mathtt{i},k})_{k\in\mathbb{N}}$
and $(\pi^{2}\nu_{\mathtt{j}|_{i_{k}}})_{k\in\mathbb{N}}$.

The following simple geometric observation justifies our choice of the roof
function.

######  Lemma 6.4.

For a.e. $(\mathtt{i},\theta)$, there exists a constant
$C(\mathtt{i},\theta)>0$ such that

$\frac{\|A_{\mathtt{i}|_{n}}|_{A_{\mathtt{i}|_{n}}^{-1}\theta}\|}{\alpha_{1}(A_{\mathtt{i}|_{n}})}\rightarrow
C(\mathtt{i},\theta)$

as $n\rightarrow\infty$.

###### Proof.

Note first that $A_{\mathtt{i}|_{n}}B(0,1)$ is an ellipse with minor axis of
length $\alpha_{1}(A_{\mathtt{i}|_{n}})$ in direction which tends to
$\theta(\mathtt{i})^{\perp}$. On the other hand,
$\|A_{\mathtt{i}|_{n}}|_{A_{\mathtt{i}|_{n}}^{-1}\theta}\|=|A_{\mathtt{i}|_{n}}B(0,1)\cap\theta|$.
Therefore, by Lemma 2.1,

$\frac{\|A_{\mathtt{i}|_{n}}|_{A_{\mathtt{i}|_{n}}^{-1}\theta}\|}{\alpha_{1}(A_{\mathtt{i}|_{n}})}=\frac{|A_{\mathtt{i}|_{n}}B(0,1)\cap\theta|}{|A_{\mathtt{i}|_{n}}B(0,1)\cap\theta(A_{\mathtt{i}|_{n}})^{\perp}|}=(1+o(n))\frac{|A_{\mathtt{i}|_{n}}B(0,1)\cap\theta|}{|A_{\mathtt{i}|_{n}}B(0,1)\cap\theta(\mathtt{i})^{\perp}|}$

and by basic geometry, the above tends to
$(\cos(d(\theta,\theta(\mathtt{i})^{\perp})))^{-1}=:C(\mathtt{i},\theta)$. ∎

In particular,

$\displaystyle d_{\rm
LP}(\mu_{M^{\ell_{k}}(\mathtt{i},\theta),kN+\log\alpha_{1}(\mathtt{i}|_{\ell_{k}})},\
\mu_{M^{\ell_{k}}(\mathtt{i},\theta),kN+\log\|A_{\mathtt{i}|_{\ell_{k}}}|_{A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta}\|-\log
C(\mathtt{i},\theta)})\rightarrow 0$

as $k\rightarrow\infty$, where the sequence
$(\ell_{k})_{k\in\mathbb{N}}=(\ell_{k}(\mathtt{i}))_{k\in\mathbb{N}}$ was
chosen to be an increasing one that tends to infinity such that

$B(\Pi(\mathtt{i}),2^{-kN})\cap\Pi(\Gamma^{\mathbb{N}})\subseteq\varphi_{\mathtt{i}|_{\ell_{k}}}(B(0,1))$

for every $k$. We will now specify our choice of the sequence. Let $N$ be
large enough so that
$2^{-N}<\min\\{d(\varphi_{i}(B(0,1)),\varphi_{j}(B(0,1))):\ i,j\in\Gamma\\}$.

Define the sequence $\ell_{k}=\ell_{k}(\mathtt{i})$ by

$\displaystyle\ell_{k}$ $\displaystyle=\max\\{n:\
-\log\|A_{\mathtt{i}|_{n}}|_{A_{\mathtt{i}|_{n}}^{-1}\theta}\|\leq(k-1)N-C(\mathtt{i},\theta)\\}$
$\displaystyle=\max\\{n:\
\|A_{\mathtt{i}|_{n}}|_{A_{\mathtt{i}|_{n}}^{-1}\theta}\|^{-1}\leq
2^{(k-1)N-\log C(\mathtt{i},\theta)}\\}$ $\displaystyle=\max\\{n:\
\alpha_{1}(A_{\mathtt{i}|_{n}})\geq 2^{-(k-1)N}\\}$

and notice for every $k$,

$B(\Pi(\mathtt{i}),2^{-kN})\cap\Pi(\Gamma^{\mathbb{N}})\subseteq\varphi_{\mathtt{i}|_{\ell_{k}}}(B(0,1)).$

With this choice,

(6.4)
$(\mathtt{i},\theta,(k-1)N-C(\mathtt{i},\theta))=(M^{\ell_{k}}(\mathtt{i},\theta),(k-1)N-C(\mathtt{i},\theta)+\log\|A_{\mathtt{i}_{\ell_{k}}}|_{A_{\mathtt{i}|_{\ell_{k}}}^{-1}\theta}\|).$

Let $F$ denote the map $Z_{\Phi}\rightarrow\mathcal{P}(\mathbb{R}^{2})$,

(6.5) $(\mathtt{i},\theta,u,t)\mapsto u\mu_{\mathtt{i},\theta,t+N}.$

######  Claim 6.5.

For every $(\mathtt{i},\theta,u)$ and $t\geq 0$,

$F(\mathtt{i},\theta,u,f(\mathtt{i},\theta)+t)=F(M(\mathtt{i},\theta),\rho(1,(\mathtt{i},\theta)u,t).$

In particular, $S_{t}\circ F=F\circ\mathcal{T}_{t}$.

###### Proof.

By the choice of $N$, we have

$B(\Pi(\mathtt{i}),2^{f(\mathtt{i},\theta)+t+N})\cap\Pi(\Gamma^{\mathbb{N}})\subseteq\varphi_{i_{0}}(B(0,1))$

for every $\mathtt{i},\theta$ and $t\geq 0$. Since
$A_{i_{0}}|_{A_{i_{0}}^{-1}\theta}=2^{-f(\mathtt{i},\theta)}R_{\theta}^{-1}\rho(1,(\mathtt{i},\theta))R_{A_{i_{0}}^{-1}\theta}$
as was noted in (5.4), it follows that

$\displaystyle F(\mathtt{i},\theta,u,f(\mathtt{i},\theta)+t)$
$\displaystyle=u\mu_{\mathtt{i},\theta,f(\mathtt{i},\theta)+t+N}$
$\displaystyle=(\varphi_{i_{0}}\mu)_{\mathtt{i},\theta,f(\mathtt{i},\theta)+t+N}$
$\displaystyle=R_{\theta}S_{f(\mathtt{i},\theta)+t+N}T_{\Pi(\mathtt{i})}(\varphi_{i_{0}}\mu)_{\Pi(\mathtt{i})}^{\theta}$
$\displaystyle=R_{\theta}S_{f(\mathtt{i},\theta)+t+N}T_{\Pi(\mathtt{i})}\varphi_{i_{0}}\mu_{\Pi(\sigma\mathtt{i})}^{A_{i_{0}}^{-1}\theta}$
$\displaystyle=\rho(1,(\mathtt{i},\theta))\mu_{M(\mathtt{i},\theta),t+N}$
$\displaystyle=F(M(\mathtt{i},\theta),\rho(1,(\mathtt{i},\theta))u,t).$

∎

By basic linear algebra,

$\|A_{i_{1}}A_{i_{2}}|_{A_{i_{2}}^{-1}A_{i_{1}}^{-1}\theta}\|=\|A_{i_{1}}|_{A_{i_{1}}^{-1}\theta}\|\cdot\|A_{i_{2}}|_{A_{i_{2}}^{-1}A_{i_{1}}^{-1}\theta}\|.$

Combining this with (6.4), we have

$F(\mathtt{i},\theta,{\rm
Id},(k-1)N-C(\mathtt{i},\theta))=\rho(\ell_{k},(\mathtt{i},\theta))\mu_{M^{\ell_{k}}(\mathtt{i},\theta),kN+\log\alpha_{1}(A_{\mathtt{i}|_{\ell_{k}}})}.$

Let $\varepsilon>0$. Then by Proposition 4.1, the above and Birkhoff’s ergodic