# On the continuity of topological entropy of certain partially hyperbolic
diffeomorphisms

Weisheng Wu  School of Mathematical Sciences, Xiamen University, Xiamen,
361005, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider certain partially hyperbolic diffeomorphisms with
center of arbitrary dimension and obtain continuity properties of the
topological entropy under $C^{1}$ perturbations. The systems considered have
subexponential growth in the center direction and uniform exponential growth
along the unstable foliation. Our result applies to partially hyperbolic
diffeomorphisms which are Lyapunov stable in the center direction. It applies
to another important class of systems which do have subexponential growth in
the center direction, for which we develop a technique to use exponential
mixing property of the systems to get uniform distribution of unstable
manifolds. A primary example is the translations on homogenous spaces which
may have center of arbitrary dimension and of polynomial orbit growth.

###### Key words and phrases:

###### 1991 Mathematics Subject Classification:

##  1\. Introduction

Entropy is one of the most important invariants of a dynamical system which
expresses the amount of “unpredictability” in the system. One of the central
subjects in dynamical systems is to investigate the robust properties of the
system under perturbations. Specifically, we are addressing in this paper the
problem of the continuity properties of the topological entropy of partially
hyperbolic diffeomorphisms under $C^{1}$ perturbations.

The concept of partial hyperbolicity was proposed independently by Brin-Pesin
and Pugh-Shub in 1970s, which is a natural generalization of uniform
hyperbolicity. The structural stability of uniformly hyperbolic
diffeomorphisms implies the topological entropy is locally constant near such
systems in the $C^{1}$ topology. For systems beyond uniform hyperbolicity, we
first briefly survey some existing results from which we can see the key
ingredients in the continuity of topological entropy.

The regularity of diffeomorphisms and the existence of homoclinic tangences
play important roles in the upper semicontinuity of the topological entropy:

  1. (1)

By [51] (also [8, 36]), the topological entropy is upper semicontinuous on the
space of $C^{\infty}$ diffeomorphisms.

  2. (2)

Misiurewicz [34] constructed $C^{r}$ counterexamples with homoclinic
tangencies for any $1\leq r<\infty$, at which the topological entropy is not
upper semicontinuous in the $C^{r}$ topology.

  3. (3)

Liao, Viana and Yang [31] proved that the topological entropy is in fact upper
semicontinuous on the space of $C^{1}$ diffeomorphisms away from tangencies.
In particular, the topological entropy is upper semicontinuous on the space of
$C^{1}$ partially hyperbolic diffeomorphisms with one-dimensional ($1$-D for
short) center.

The lower semicontinuity of the topological entropy is usually due to the
existence of certain types of dynamical structure which are robust and carry
entropy:

  1. (1)

It is well known that horseshoes carry positive entropy and they are
persistent after small perturbations. Then by a classical result of Katok [26]
the topological entropy is lower semicontinuous on the space of $C^{1+\alpha}$
diffeomorphisms on surfaces.

  2. (2)

In [23], the authors considered a $C^{1}$ partially hyperbolic diffeomorphism
$f$ whose stable and unstable foliations stably carry some unique non-trivial
homologies. Then if $f$ has $1$-D center, the topological entropy is constant
in a small $C^{1}$ neighborhood of $f$; if $f$ has $2$-D center, the
topological entropy is continuous at $f$ in the $C^{\infty}$ topology111By
[23], under the assumption that the unstable foliation of $f$ stably carries a
unique non-trivial homology, the volume growth along unstable foliation
$\chi_{u}(f)$ is locally constant at $f$. $\chi_{u}(f)$ can also be understood
as the unstable topological entropy of $f$ by [20]..

  3. (3)

In [46], the authors proved for $f$ which is any small $C^{1}$ perturbation of
a time-one map of a hyperbolic flow, the local unstable manifolds have uniform
rate of expansion, which implies the (unstable) topological entropy is lower
semicontinuous at $f$ in the $C^{1}$ topology. Since center bundle is one
dimensional, the topological entropy is continuous at $f$ in the $C^{1}$
topology.

Please see also the papers [45, 11, 9] for recent progress.

Let us focus on the topological entropy of $C^{1}$ partially hyperbolic
diffeomorphisms. As one can see from above, most known results are established
for partially hyperbolic diffeomorphisms with $1$-D or $2$-D center. In [46],
the authors conjectured the topological entropy is continuous on the space of
$C^{1}$ partially hyperbolic diffeomorphisms with $1$-D center. Indeed, the
conjecture is true for all the known examples. In contrast, when center has
dimension two, the topological entropy can fail to be continuous; such an
example is constructed in [23]. Then it is natural to investigate the
topological entropy for more partially hyperbolic diffeomorphisms with higher
dimensional center.

Partial hyperbolicity includes the hyperbolic part and the center part. In
[20], the authors introduced the notion of unstable topological entropy, which
is caused by the hyperbolic part of the system. Recently, the unstable metric
entropy is studied in [20] and [50], see also the survey [47]. In this paper,
we focus on $C^{1}$ partially hyperbolic diffeomorphisms with subexponential
growth in the center direction. Then only the hyperbolic part contributes to
the topological entropy, which can be verified using the theory of unstable
entropy. We remark that Ledrappier-Young formula which holds only for
$C^{r},r>1$ diffeomorphisms, is not directly applied here.

On the other hand, we investigate the continuity of unstable topological
entropy in the $C^{1}$ topology. The upper semicontinuity turns out to be a
general result for all $C^{1}$ partially hyperbolic diffeomorphisms. The lower
semicontinuity is more delicate, and we obtain it under the condition that the
unstable manifolds have uniform exponential expansion, which has appeared in
[46].

As explained above, our main result in this paper states that the topological
entropy function is lower semicontinuous at partially hyperbolic
diffeomorphisms which have subexponential growth in the center direction and
uniform exponential growth along the unstable foliation. This result applies
to partially hyperbolic diffeomorphisms which is topologically transitive and
Lyapunov stable in the center direction (hence to those with bounded expansion
in the center direction). This class includes time-one maps of frame flows;
semisimple translations on homogenous spaces; time-one map of geodesic flows
in symmetric spaces of noncompact type, etc. An interesting phenomenon is that
Lyapunov stable center together with topological transitivity implies uniform
exponential growth along the unstable foliation.

A more difficult situation is when the system does have subexponential growth
in center direction (for example, polynomial growth). Our technique is to
establish effective density of the unstable foliation under the assumption
that the system has an exponential mixing property. The fast equidistribution
of the unstable foliation then eliminates the effect caused by subexponential
growth in the center direction, which guarantees the uniform exponential
growth along the unstable foliation. An important example would be the
translations on homogeneous spaces. Let $G$ be a connected semisimple Lie
group, $\Gamma$ a cocompact lattice in $G$, and denote by $T_{g}$ the map
given by the left translation on $G/\Gamma$ by $g\in G$. When $\mathrm{Ad}g$
is non-quasiunipotent, the system $(G/\Gamma,g)$ is a primary example of
partially hyperbolic diffeomorphisms (cf. Theorem D in [40]). The system may
have polynomial growth in the center direction, and indeed it satisfies a
version of exponential mixing property (“Condition (EM)” as in [28, Section
2.4.1]). It is also interesting to consider $C^{1}$ perturbations of $T_{g}$
as well as the time one map of a smooth time change of a homogenous flow. We
do not know if such systems have certain exponential mixing property. At last,
similar results are obtained for ergodic toral automorphisms, via effective
density of their unstable foliations.

###  1.1. Statement of main results

In general, for a compact metric space $(X,d)$, a continuous map
$T:X\rightarrow X$ and $\forall~{}n\in{\mathbb{N}}$, we define a new distance
$d_{n}$ on $X$ by

$\displaystyle d_{n}:X~{}\times~{}X$ $\displaystyle\rightarrow$
$\displaystyle\mathbb{R}$ $\displaystyle(x,y)$ $\displaystyle\mapsto$
$\displaystyle d_{n}(x,y)=\max_{0\leq j<n}d(T^{j}(x),T^{j}(y)),$

which measures the maximal distance between the two orbits of $x$ and $y$ up
to time $n$. The new metric space $(X,d_{n})$ is obviously homeomorphic to
$(X,d)$.

For any $\epsilon>0$, we say that a subset $A\subset X$ is an
$(n,\epsilon)$-separated set, if for any distinct points $x,y\in A$,
$d_{n}(x,y)>\epsilon$. The compactness of $X$ implies that an
$(n,\epsilon)$-separated set is a finite set. Let

$N(T,n,\epsilon):=\max\\{\\#A\ |A\ \text{ is
an}~{}(n,\epsilon)\text{-separated set in\ }X\\}$

where $\\#A$ denotes the cardinality of $A$. The topological entropy of the
map $T$ is defined by the following formula:

$h_{{\mathop{\hbox{\footnotesize\rm top}}}}(T):=\lim_{\epsilon\rightarrow
0^{+}}\limsup_{n\to\infty}\frac{1}{n}\log N(T,n,\epsilon).$

Topological entropy can be defined equivalently by using
$(n,\epsilon)$-spanning sets or open covers of $X$. See Chapter $7$ in [48] or
Chapter $3$ in [27] for comprehensive discussions on topological entropy.

Let $M$ be a $d$-dimensional smooth, closed Riemannian manifold and let
$f:M\to M$ be a $C^{1}$ diffeomorphism. $f$ is called a _partially hyperbolic
diffeomorphism_ (abbreviated as _PHD_) if there exists a nontrivial
$Df$-invariant splitting of the tangent bundle $TM=E_{f}^{s}\oplus
E_{f}^{c}\oplus E_{f}^{u}$ into so-called stable, center, and unstable
distributions, such that all unit vectors $v^{\sigma}\in E_{f}^{\sigma}(x)$
($\sigma=s,c,u$) with $x\in M$ satisfy

$\|D_{x}fv^{s}\|<\|D_{x}fv^{c}\|<\|D_{x}fv^{u}\|,$

and

$\|D_{x}f|_{E_{f}^{s}(x)}\|<1,\ \ \ \text{\ and\ \ \ \
}\|D_{x}f^{-1}|_{E_{f}^{u}(x)}\|<1,$

for some suitable Riemannian metric on $M$. The stable distribution
$E_{f}^{s}$ and unstable distribution $E_{f}^{u}$ are integrable: there exist
so-called stable and unstable foliations $W_{f}^{s}$ and $W_{f}^{u}$
respectively such that $TW_{f}^{s}=E_{f}^{s}$ and $TW_{f}^{u}=E_{f}^{u}$. We
say that $f$ is _dynamically coherent_ if there exist $f$-invariant center-
stable and center-unstable foliations $W_{f}^{cs}$ and $W_{f}^{cu}$ tangent to
$E_{f}^{cs}$ and $E_{f}^{cu}$ respectively; then there exists a center
foliation $W_{f}^{c}$ tangent to $E_{f}^{c}$ by intersecting leaves of
$W_{f}^{cs}$ and $W_{f}^{cu}$. See for example [43] for the basic properties
of PHDs. We mention that $C^{r}$ PHDs form an open subset of $C^{r}(M)$, the
space of $C^{r}$ diffeomorphims of $M$ for any $1\leq r\leq\infty$.

We will recall the definitions of unstable entropy
$h^{u}_{{\mathop{\hbox{\footnotesize\rm top}}}}(f)$ in Section 2. Our first
result below is for general PHDs.

######  Proposition 1.1.

Let $f:M\to M$ be any $C^{1}$ partially hyperbolic diffeomorphism. Then

$h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$

is upper semicontinuous at $f$.

Note that in a recent paper [45], the authors have shown that the function of
unstable volume growth is upper semicontinuous in the $C^{1}$ topology using
geometric arguments. And it is already proved in [20] that unstable
topological entropy and unstable volume growth coincide. Our proof of
Proposition 1.1 uses unstable metric entropy and its upper semicontinuity.
Modulo Proposition 1.1, the main work in this paper is to study whether
unstable topological entropy is lower semicontinuous.

We focus on a $C^{1}$ PHD $f:M\to M$ _with subexponential growth in the center
direction_ , that is, for any $\epsilon>0$, there exists a constant
$C_{\epsilon}>0$ such that $\|D_{x}f^{n}|_{E_{f}^{c}(x)}\|\leq
C_{\epsilon}e^{n\epsilon}$ for any $n\in\mathbb{N}$ and any $x\in M$ (cf.
[6]). As for the unstable direction, we introduce the following terminology in
order to simplify the presentation.

######  Definition 1.2.

We say that a $C^{1}$ PHD $f:M\to M$ has _uniform exponential growth along the
unstable foliation_ if for any $\rho>0$ and $\delta>0$ small enough, there
exists $N(\rho,\delta)\in{\mathbb{N}}$ such that for any $x\in M$,
$f^{N(\rho,\delta)}W^{u}_{f}(x,\delta)$ contains at least
$e^{N(\rho,\delta)(h-3\rho)}$ disjoint sets of the form
$\\{W^{u}_{f}(z_{i},2\delta):z_{i}\in M\\}$, where
$h=h_{\mathop{\hbox{\footnotesize\rm top}}}^{u}(f)$ and $W^{u}_{f}(x,\delta)$
denotes the open ball centered at $x$ of radius $\delta$ with respect to the
metric $d^{u}$ inherited from the Riemannian structure on $W^{u}(x)$.

Our main result in this paper is the following:

######  Theorem A.

Let $f:M\to M$ be a $C^{1}$ partially hyperbolic diffeomorphism.

  1. (1)

If $f$ has uniform exponential growth along the unstable foliation, then the
unstable topological entropy function $h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at $f$.

  2. (2)

Assume further that $f$ has subexponential growth in the center direction,
then the topological entropy function $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is lower semicontinuous at
$f$. If in addition $f$ is $C^{\infty}$, $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{\infty}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at $f$.

We apply Theorem A to several important classes of PHDs.

A PHD is called _Lyapunov stable in the center direction (or just Lyapunov
stable)_ if given any $\epsilon>0$, one can find $\delta>0$ such that for any
$C^{1}$-arc $\gamma$ tangent to $E^{c}$ with length less than $\delta$, the
curves $f^{n}\gamma,n\in\mathbb{N}$ all have length less than $\epsilon$ (cf.
[42]). When $f$ and $f^{-1}$ are both Lyapunov stable, we say that $f$ is bi-
Lyapunov stable. It is known that if $f$ is Lyapunov stable then $E^{c}\oplus
E^{s}$ integrates to a unique $f$-invariant foliation $W^{cs}$. In particular,
if $f$ is bi-Lyapunov stable, then $f$ is dynamically coherent (cf. Theorem
7.5 in [42]).

A $C^{1}$ PHD $f:M\to M$ is said to have _bounded expansion in the center
direction_ if there exists $K>1$ such that

$\|Df^{n}(x)|_{E_{f}^{c}(x)}\|<K,\quad\forall n\in\mathbb{N}.$

It is clear that bounded expansion implies both Lyapunov stability and
subexponential growth in the center direction.

######  Theorem B.

Let $f:M\to M$ be a topologically transitive $C^{1}$ partially hyperbolic
diffeomorphism which is Lyapunov stable in the center direction. Then
$h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at $f$. If in
addition $f$ has bounded expansion in the center direction, then the
topological entropy function $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is lower semicontinuous at
$f$.

Our technique to prove Theorem B, particularly the uniform exponential growth
of $f$ along the unstable foliation, is to use effective density of the
unstable foliation of $f$. The technique applies to another important class of
PHDs which we describe below.

Consider a volume preserving PHD $f:M\to M$, i.e., $f$ preserves a smooth
probability measure $m$ on $M$. Denote by $m_{x}^{u}$ the volume induced by
the Riemannian structure on $W^{u}(x)$. For any
$r\in{\mathbb{N}}\cup\\{\infty\\}$, let $C^{r}(M,{\mathbb{R}})$ denote the set
of all the $C^{r}$ functions from $M$ to ${\mathbb{R}}$ equipped with the
$C^{r}$-norm $\|\cdot\|_{r}$. We say $f$ has _property (EM)_ , if there exists
$r\in{\mathbb{N}}$, $C>0$, $\alpha>0$ and $\delta_{0}>0$, such that for any
$0<\delta\leq\delta_{0}$, any $\phi,\psi\in C^{r}(M,{\mathbb{R}})$,

(1)
$\displaystyle\left|\int_{W_{f}^{u}(x,\delta)}\phi(f^{n}(p))\psi(p)dm_{x}^{u}(p)-\int\phi
dm\int_{W_{f}^{u}(x,\delta)}\psi
dm_{x}^{u}\right|<C\|\phi\|_{r}\|\psi\|_{r}e^{-n\alpha}$

for any $n\in\mathbb{N}$ and any $x\in M$. See [14] for a related notion of
exponential mixing.

######  Theorem C.

Let $f:M\to M$ be a $C^{1}$ dynamically coherent partially hyperbolic
diffeomorphism which has subexponential growth in the center direction and
satisfies property (EM). Then the topological entropy function
$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is lower semicontinuous at
$f$. Furthermore, $h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at $f$.

An important example of PHDs with property (EM) is the partially hyperbolic
translations on compact homogeneous spaces. Let $G$ be a connected semisimple
Lie group and $\Gamma\subset G$ be a cocompact lattice. Choose and fix a norm
on the Lie algebra $\mathfrak{g}$ of $G$, which induces a right invariant
Riemannian metric “dist” on $G$. Then set

$d(g\Gamma,h\Gamma):=\inf_{\gamma\in\Gamma}\mathrm{dist}(g,h\gamma).$

We study the topological entropy of the dynamical system $T_{g}:G/\Gamma\to
G/\Gamma$ induced by the left translation of $T_{g}(x)=gx$, for any $x\in
G/\Gamma$. Note that $T_{g}:G/\Gamma\to G/\Gamma$ is a $C^{\infty}$ PHD when
$\mathrm{Ad}g$ is nonquasiunipotent, see Section $5.2$ for more details. The
polynomial orbit growth in the center direction causes substantial
difficulties in many important problems in homogenous dynamics (see for
example [28, 25]). The following theorem is a corollary of Theorem C.

######  Theorem D.

Let $G$ be a connected semisimple Lie group, $\Gamma$ be a cocompact lattice
of $G$, and $g\in G$. Then the topological entropy function
$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(G/\Gamma)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is lower semicontinuous
at $T_{g}$ and $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{\infty}(G/\Gamma)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at
$T_{g}$.

######  Remark 1.3.

The term “Condition (EM)” is used in [28, Section 2.4.1] for
$(G/\Gamma,g^{t})$ where $g^{t}$ is a one-parameter subgroup of $G$ (other
conditions on $G/\Gamma$ are imposed), from which our property (EM) (1) can be
derived (cf. [28, Proposition 2.4.8]). In our discrete setting, property (EM)
follows from [15, Propositions 3.1] (see Section 5.2). It is interesting to
know whether the time-one map of a smooth time change of $(G/\Gamma,g^{t})$
satisfies property (EM).

######  Theorem E.

Let $T_{A}:\mathbb{T}^{d}\to\mathbb{T}^{d}$ be an ergodic toral automorphism.
Then the topological entropy function $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(\mathbb{T}^{d})\to{\mathbb{R}}^{+}\cup\\{0\\}$ is lower
semicontinuous at $T_{A}$. Furthermore,
$h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(\mathbb{T}^{d})\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at
$T_{A}$, and $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{\infty}(\mathbb{T}^{d})\to{\mathbb{R}}^{+}\cup\\{0\\}$ is
continuous at $T_{A}$.

######  Remark 1.4.

We will state Theorem D’, an improvement of Theorem D in Section 5.2, and give
a simpler proof. Similarly, Theorem E can also be improved, see Theorem E’ in
Section 5.3. Nevertheless, our technique provides a systematic way to treat
these problems.

The paper is organized as follows. In Section $2$, we recall the notion of
unstable entropy and prove Proposition 1.1. Section 3 is devoted to proving
main Theorem A. In section 4, we consider PHDs with Lyapunov stable center and
prove Theorem B. Theorems C, D and E are proved in the last section, where we
also summarize the known examples of PHDs at which the topological entropy has
continuity properties.

##  2\. Unstable topological entropy

In this section, we collect some results on unstable topological entropy, and
prove Proposition 1.1. The results in this section except Proposition 2.6 hold
for general PHDs.

Let $M$ be a $d$-dimensional smooth, closed manifold and let $f:M\to M$ be a
$C^{1}$ PHD. We start to recall the definition of unstable topological entropy
given in [20].

Denote by $d^{\sigma}$ the metric induced by the Riemannian structure on
$W^{\sigma}_{f}$, $\sigma\in\\{s,u,cs,cu,c\\}$ if the foliation exists, and
then denote by $W_{f}^{\sigma}(x,\delta)$ the open ball inside
$W_{f}^{\sigma}(x)$ centered at $x\in M$ of radius $\delta>0$ with respect to
the metric $d^{\sigma}$. Given any $y\in W_{f}^{u}(x)$, define

$d^{u}_{n}(x,y):=\max_{0\leq j<n}d^{u}(f^{j}(x),f^{j}(y)).$

Let $N^{u}(f,n,\epsilon,x,\delta)$ be the maximal number of points in
$\overline{W_{f}^{u}(x,\delta)}$ with pairwise $d^{u}_{n}$-distances at least
$\epsilon$. We call such set a maximal _$(n,\epsilon)$ u-separated set_ of
$\overline{W_{f}^{u}(x,\delta)}$.

######  Definition 2.1.

The _unstable topological entropy_ of $f$ on $M$ is defined by

$h^{u}_{\text{top}}(f):=\lim_{\delta\to 0}\sup_{x\in
M}h_{\text{top}}(f,\overline{W_{f}^{u}(x,\delta)}),$

where

$h_{\text{top}}(f,\overline{W_{f}^{u}(x,\delta)}):=\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log N^{u}(f,n,\epsilon,x,\delta).$

######  Remark 2.2.

One can also use $(n,\epsilon)$ u-spanning sets or open covers to give
equivalent definitions for unstable topological entropy. Moreover,

$h^{u}_{\text{top}}(f)=\sup_{x\in
M}h_{\text{top}}(f,\overline{W_{f}^{u}(x,\delta)})$

for any $\delta>0$ small enough (cf. Lemma 4.1 in [20]).

The unstable volume growth (cf. [23]) is defined as follows:

$\chi_{u}(f):=\sup_{x\in M}\chi_{u}(x,\delta)$

where

$\chi_{u}(x,\delta):=\limsup_{n\to\infty}\frac{1}{n}\log(\text{Vol}(f^{n}(W_{f}^{u}(x,\delta))).$

It is independent of the choice of $\delta>0$ and the Riemannian metric. As
mentioned in the introduction, by Theorem C in [20],

$h^{u}_{\text{top}}(f)=\chi_{u}(f).$

Let $\mu$ be an $f$-invariant Borel probability measure on $M$. In [20], the
authors give a new definition for the unstable metric entropy $h_{\mu}^{u}(f)$
which coincides with the one considered by Ledrappier and Young [30], by using
measurable partitions subordinate to unstable manifolds that can be obtained
by refining a finite Borel partition into pieces of unstable leaves. We skip
this definition here, just recalling two important properties of unstable
metric entropy as follows.

Let $\mathcal{M}_{f}(M)$ (resp. $\mathcal{M}^{e}_{f}(M)$) denote the set of
all $f$-invariant (resp. $f$-ergodic) Borel probability measures on $M$. The
following variational principle is proved in Theorem D of [20].

######  Theorem 2.3.

Let $f:M\to M$ be any $C^{1}$ PHD. Then

$h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f)=\max_{\mu\in\mathcal{M}_{f}(M)}h_{\mu}^{u}(f)=\max_{\mu\in\mathcal{M}^{e}_{f}(M)}h_{\mu}^{u}(f).$

The upper semicontinuity of the unstable metric entropy function

$h^{u}_{\\{\cdot\\}}(f):{\mathcal{M}}_{f}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$

is proved in Proposition 2.15 in [20]. In fact, Theorem A in [50] gives the
following more general result:

######  Theorem 2.4.

Let $f_{n},f:M\to M$ be $C^{1}$ PHDs such that $f_{n}\to f$ in the $C^{1}$
topology as $n\to\infty$. Suppose that $\mu_{n}\in{\mathcal{M}}_{f_{n}}(M)$,
and $\mu_{n}\to\mu$ in the weak∗ topology as $n\to\infty$. Then

$\limsup_{n\to\infty}h^{u}_{\mu_{n}}(f_{n})\leq h^{u}_{\mu}(f).$

We are ready to prove Proposition 1.1, that is,
$h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is upper semicontinuous at any
$C^{1}$ PHD $f$.

###### Proof of Proposition 1.1.

For any $C^{1}$ PHD $f:M\to M$, let $f_{n}\to f$ in the $C^{1}$ topology as
$n\to\infty$. Then $f_{n}$ are also PHDs. According to Theorem 2.3, we know
that for each $f_{n}$, there exists $\mu_{n}\in{\mathcal{M}}_{f_{n}}(M)$ such
that

(2) $h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f_{n})=h^{u}_{\mu_{n}}(f_{n}).$

Let $n_{k}$ be a subsequence such that

(3) $\limsup_{n\to\infty}h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f_{n})=\lim_{k\to\infty}h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f_{n_{k}}).$

Then a subsequence of $\mu_{n_{k}}$ converges to a probability measure $\mu$
on $M$. Without loss of generality, we assume $\mu_{n_{k}}\to\mu$. It is
straightforward to verify that $\mu\in{\mathcal{M}}_{f}(M)$. By Theorem 2.4,
we have

(4) $\lim_{k\to\infty}h^{u}_{\mu_{n_{k}}}(f_{n_{k}})\leq h^{u}_{\mu}(f).$

By Theorem 2.3, $h^{u}_{\mu}(f)\leq h^{u}_{\mathop{\hbox{\footnotesize\rm
top}}}(f)$. Thus by (2),(3) and (4),

$\limsup_{n\to\infty}h^{u}_{{\mathop{\hbox{\footnotesize\rm top}}}}(f_{n})\leq
h^{u}_{\mathop{\hbox{\footnotesize\rm top}}}(f).$

The proof of the proposition is complete. ∎

Let us consider a $C^{1}$ PHD $f:M\to M$ with subexponential growth in the
center direction, that is, for any $\epsilon>0$, there exists a constant
$C_{\epsilon}>0$ such that $\|D_{x}f^{n}|_{E_{f}^{c}(x)}\|\leq
C_{\epsilon}e^{n\epsilon}$ for any $n\in\mathbb{N}$ and any $x\in M$. The
following lemma is useful for $C^{1}$ PHDs.

######  Lemma 2.5.

(Cf. [23]) Let $f:M\to M$ be a $C^{1}$ PHD and $\mu$ an $f$-invariant and
ergodic measure. Then

$h_{\mu}(f)\leq\chi^{u}(f)+\sum_{\lambda_{i}^{c}(\mu)>0}m_{i}(\mu)\lambda_{i}^{c}(\mu)$

where the sum is taken over all the positive Lypapunov exponents
$\lambda_{i}^{c}(\mu)$ of $\mu$ in the center direction with multiplicity
$m_{i}(\mu)$.

######  Proposition 2.6.

If a $C^{1}$ PHD $f:M\to M$ has subexponential growth in the center direction,
then $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f)=h^{u}_{{\mathop{\hbox{\footnotesize\rm top}}}}(f)$.

###### Proof.

Let $\nu$ be any $f$-invariant and ergodic Borel probability measure on $M$.
It is clear that the Lyapunov exponents of $\nu$ in the center direction are
always nonpositive, since $f$ has subexponential growth in the center
direction. Thus by Lemma 2.5 and the fact $h^{u}_{\text{top}}(f)=\chi_{u}(f)$,
we have $h_{\nu}(f)\leq h^{u}_{\text{top}}(f)$. Thus by the variational
principle, we have

$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f)=\sup_{\nu\in\mathcal{M}^{e}_{f}(M)}h_{\nu}(f)\leq
h^{u}_{{\mathop{\hbox{\footnotesize\rm top}}}}(f).$

$h^{u}_{{\mathop{\hbox{\footnotesize\rm top}}}}(f)\leq
h_{{\mathop{\hbox{\footnotesize\rm top}}}}(f)$ is automatic and therefore we
finish the proof of the proposition. ∎

######  Remark 2.7.

If $f:M\to M$ is a $C^{r}$-PHD, $r>1$, we can also obtain
$h_{\nu}(f)=h^{u}_{\nu}(f)$ and $h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f)=h_{{\mathop{\hbox{\footnotesize\rm top}}}}(f)$, by using
Ledrappier-Young formula (cf. [30]).

######  Remark 2.8.

A $C^{1}$ PHD $f:M\to M$ has subexponential growth in the center direction if
and only if the Lyapunov exponents in the center direction are all nonpositive
with respect to any $f$-invariant measure. The “only if” part is clear. For
the “if” part, assume that for some $\epsilon>0$, there exists a sequence of
points $x_{n}\in M$ such that $\|D_{x_{n}}f^{n}|_{E_{f}^{c}(x_{n})}\|\geq
e^{n\epsilon}$ for $n\geq 0$. Then a subsequence of the measures

$\mu_{n}:=\frac{1}{n}\sum_{i=0}^{n}f^{i}_{*}\delta_{x_{n}}$

converges to an $f$-invariant measure $\mu$, which has some positive central
Lyapunov exponents, so we get a contradiction.

##  3\. Proof of Theorem A

In this section we prove Theorem A. Recall that a $C^{1}$ PHD $f:M\to M$ has
uniform exponential growth along the unstable foliation, if for any $\rho>0$
and any $\delta>0$ small enough, there exists $N(\rho,\delta)\in{\mathbb{N}}$
such that for any $x\in M$, $f^{N(\rho,\delta)}W^{u}_{f}(x,\delta)$ contains
at least $e^{N(\rho,\delta)(h-3\rho)}$ disjoint sets of the form
$\\{W^{u}_{f}(z_{i},2\delta):z_{i}\in M\\}$, where
$h=h_{\mathop{\hbox{\footnotesize\rm top}}}^{u}(f)$.

Let $D^{k}$ denote the unit open disk in ${\mathbb{R}}^{k}$ and $d=\dim M$.
Let $\mathcal{F}$ be a continuous foliation of $M$ with $C^{1}$ leaves, then
an _$\mathcal{F}$ -foliation box_ is the image $B$ of a topological embedding
$\Phi:D^{d-l}\times D^{l}\to M$ which sends each plaque $\Phi(\\{x\\}\times
D^{l})$ to a leaf of $\mathcal{F}$. Each $\Phi(x,\cdot):D^{l}\to
M,y\mapsto\Phi(x,y)$ is a $C^{1}$ embedding depending continuously on $x$ in
the $C^{1}$ topology. We write this $\mathcal{F}$-foliation box by
$\\{B,\Phi,D\\}$ where $D=\Phi(D^{d-l}\times\\{0\\})$.

######  Proposition 3.1.

Let $f:M\to M$ be a $C^{1}$ PHD with uniform exponential growth along the
unstable foliation. Then for any given $\rho>0$, there exists a $C^{1}$ open
neighborhood $\mathcal{U}_{f}(\rho)$ of $f$ such that for any
$g\in\mathcal{U}_{f}(\rho)$ and any $x\in M$,
$g^{N(\rho,\delta)}W^{u}_{g}(x,\delta)$ contains at least
$e^{N(\rho,\delta)(h-3\rho)}$ disjoint sets of the form
$W^{u}_{g}(z_{i},\delta)$ with $z_{i}\in M$.

###### Proof.

If $f_{n}\to f$ in the $C^{1}$ topology, then the foliations $W^{u}_{f_{n}}$
converge to $W_{f}^{u}$ in the following sense:

  1. (1)

For each $n$, there exist finite covers of $M$ by $W^{u}_{f_{n}}$\- and
$W^{u}_{f}$-foliation boxes

$\\{B^{n}_{i},\Phi^{n}_{i},D_{i}\\}^{k}_{i=1}\quad\text{and\quad}\\{B_{i},\Phi_{i},D_{i}\\}^{k}_{i=1}$

respectively;

  2. (2)

for each $1\leq i\leq k$, the topological embeddings

$\Phi^{n}_{i}:D^{d-l}\times D^{l}\to M$

converge uniformly to $\Phi_{i}$ in the $C^{0}$ topology as $n\to\infty$;

  3. (3)

for every $1\leq i\leq k$ and $x\in D_{i},\Phi_{i}^{n}(x,\cdot):D^{l}\to M$
defined by

$y\mapsto\Phi_{i}^{n}(x,y)$

are $C^{1}$ embeddings converging to $\Phi_{i}(x,\cdot)$ in the $C^{1}$
topology as $n\to\infty$.

Now $f$ has uniform exponential growth along the unstable foliation. Without
loss of generality, one can choose $\delta>0$ to be much smaller than the
Lebesgue number of the open cover $\\{B_{i}\\}_{i=1}^{k}$ of $M$. Since
$N(\rho,\delta)$ is already fixed, the conclusion of the proposition follows
from the above continuity of unstable foliations and the uniform exponential
growth of $f$ along the unstable foliation. ∎

###### Proof of Theorem A.

Assume that $f$ has uniform exponential growth along the unstable foliation.
By Proposition 3.1, for any $\rho>0$, there exists a $C^{1}$ open neighborhood
$\mathcal{U}_{f}(\rho)$ of $f$ such that for any $g\in\mathcal{U}_{f}(\rho)$,
and for any $x\in M$, $g^{N(\rho,\delta)}W^{u}_{g}(x,\delta)$ contains at
least $e^{N(\rho,\delta)(h-3\rho)}$ disjoint sets of the form
$W^{u}_{g}(z_{i},\delta)$ with $z_{i}\in M.$ By induction, for any
$l\in{\mathbb{N}}$, $g^{lN(\rho,\delta)}W^{u}_{g}(x,\delta)$ contains at least
$e^{lN(\rho,\delta)(h-3\rho)}$ disjoint sets of the form
$W^{u}_{g}(w_{i},\delta)$ with $w_{i}\in M.$ Then
$\\{g^{-lN(\rho,\delta)}w_{i}\\}$ is a $(lN(\rho,\delta),\delta)$ u-separated
subset of $W^{u}_{g}(x,\delta)$. So

$N^{u}(g,lN(\rho,\delta),\delta,x_{0},\delta)\geq
e^{lN(\rho,\delta)(h-3\rho)},$

which implies

(5) $h_{{\mathop{\hbox{\footnotesize\rm top}}}}(g)\geq
h^{u}_{{\mathop{\hbox{\footnotesize\rm top}}}}(g)\geq h-3\rho.$

This gives the lower semicontinuity in $C^{1}$ topology of the function
$h^{u}_{{\mathop{\hbox{\footnotesize\rm top}}}}(\cdot)$ at $f$. The upper
semicontinuity follows from Proposition 1.1 and hence
$h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at $f$. This
proves Theorem A(1).

If $f$ also has subexponential growth in the center direction, then
$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f)=h^{u}_{{\mathop{\hbox{\footnotesize\rm top}}}}(f)$ by Proposition
2.6. So the topological entropy function $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is lower semicontinuous at $f$
by (5). If in addition $f$ is $C^{\infty}$, the upper semicontinuity of
$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{\infty}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is a classical result
(cf. for example [36]). Therefore, $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{\infty}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at $f$. The
proof of Theorem A is complete. ∎

##  4\. Lyapunov stable PHDs

Recall that a PHD is called Lyapunov stable (in center direction) if given any
$\epsilon>0$, one can find $\delta>0$ such that for any $C^{1}$-arc $\gamma$
tangent to $E^{c}$ with length less than $\delta$, the curves
$f^{n}\gamma,n\in\mathbb{N}$ all have length less than $\epsilon$. If $f$ is
Lyapunov stable then $E^{c}\oplus E^{s}$ integrates to a unique $f$-invariant
foliation $W^{cs}$ (cf. [42]).

A $C^{1}$ PHD $f:M\to M$ is said to have bounded expansion in the center
direction if there exists $K>1$ such that

$\|Df^{n}(x)|_{E_{f}^{c}(x)}\|<K,\quad\forall n\in\mathbb{N}.$

Bounded expansion implies both Lyapunov stability and subexponential growth in
the center direction.

###  4.1. Proof of Theorem B

We will show that a topologically transitive and Lyapunov stable $C^{1}$ PHD
$f:M\to M$ has uniform exponential growth along unstable manifolds, and hence
Theorem A(1) applies. If further, $f$ has bounded expansion in the center
direction, then $f$ has subexponential growth in center direction and Theorem
A(2) applies.

The following property is a version of effective density of the unstable
foliation.

######  Lemma 4.1.

(Effective density, cf. [10, Lemma 6.2]) Let $f:M\to M$ be a $C^{1}$ PHD which
is topologically transitive and Lyapunov stable. Then given any $\delta>0$,
there is $n\in{\mathbb{N}}$ such that for every $x,y\in M$, there is $0\leq
k\leq n$ such that

$f^{k}(W_{f}^{u}(x,\delta))\cap W_{f}^{cs}(y,\delta)\neq\emptyset.$

######  Proposition 4.2.

Let $f:M\to M$ be a $C^{1}$ PHD which is topologically transitive and Lyapunov
stable. Then $f$ has uniform exponential growth along unstable foliation.

###### Proof.

Let $h=h_{\mathop{\hbox{\footnotesize\rm top}}}^{u}(f)$ and $\delta_{0}>0$ be
small enough to be specified later. Recall the definition of unstable
topological entropy and Remark 2.2. For any $\rho>0$, we can find $x_{0}\in M$
and $0<\delta\ll\delta_{0}$ small enough such that

$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f,\overline{W^{u}_{f}(x_{0},\delta)})>h-\rho,$

and if $d^{cs}(x,y)<\delta$ then $d^{cs}(f^{n}x,f^{n}y)\ll\delta_{0}/2$ by
Lyapunov stability of $f$. Let $0<\epsilon\ll\delta$ be small enough and
$N_{1}\in{\mathbb{N}}$ large enough such that if $n\geq N_{1}$, then there
exists an $(n,\epsilon)$ u-separated subset $S$ of
$\overline{W^{u}_{f}(x_{0},\delta)}$ such that
$\\#S=N^{u}(f,n,\epsilon,x_{0},\delta)>e^{n(h-2\rho)}$.

Since $f$ is Lyapunov stable, the center-stable foliation $W_{f}^{cs}$ exists
and it is continuous. Shrinking $\delta_{0}$ if necessary, for any $y\in
W^{cs}_{f}(x,\delta_{0})$ and any $0<\delta<\delta_{0}$, the $cs$-holonomy map
$H_{x,y}^{cs}$ is well defined, and such that

$H^{cs}_{x,y}(\overline{W^{u}_{f}(x,\delta)})\subset
W^{u}_{f}(y,c_{2}(\delta))$

and

$H^{cs}_{x,y}(W^{u}_{f}(x,\epsilon/2))\supset
W^{u}_{f}(y,c_{3}(\epsilon,\delta))$

for some constants $c_{2}=c_{2}(\delta)>2\delta$ and
$c_{3}=c_{3}(\epsilon,\delta)>0$. Then take $N_{3}\in{\mathbb{N}}$ large
enough such that for any $n\geq N_{3}$ and any $x\in M$:

  1. (1)

$f^{n}(W^{u}_{f}(x,c_{3}))\supset W^{u}_{f}(f^{n}(x),2\delta)$;

  2. (2)

$f^{n}(W^{u}_{f}(x,\delta))\supset W^{u}_{f}(f^{n}(x),2c_{2})$.

Now take any $x\in M$. Applying Lemma 4.1 with $x=f^{N_{3}}x$ and $y=x_{0}$,
there exists $N_{2}\in{\mathbb{N}}$ depending only on $\delta$ such that

$f^{k}(W_{f}^{u}(f^{N_{3}}x,\delta))\cap
W_{f}^{cs}(x_{0},\delta)\neq\emptyset$

for some $0\leq k\leq N_{2}$. Then $f^{k+N_{3}}(W_{f}^{u}(x,\delta))\cap
W_{f}^{cs}(x_{0},\delta)\neq\emptyset$. Let

$y\in f^{k+N_{3}}(W_{f}^{u}(x,\delta))\cap W_{f}^{cs}(x_{0},\delta),$

then we have $f^{k+N_{3}}(W_{f}^{u}(x,\delta))\supset W^{u}_{f}(y,c_{2})$ by
$c_{2}>2\delta$ and the choice of $N_{3}$. So

$H_{x_{0},y}^{cs}(\overline{W^{u}_{f}(x_{0},\delta)})\subset
f^{k+N_{3}}(W_{f}^{u}(x,\delta)).$

Note that $k$ is depending on $x,x_{0}$, but we shall use $N_{2}$ instead in
the definition of $N(\rho,\delta)$ later.

Finally, we take $n_{\rho}\in{\mathbb{N}}$ large enough such that
$n_{\rho}>\max\\{N_{1},N_{2},N_{3}\\}$ and

$\frac{n_{\rho}}{2N_{3}+N_{2}+n_{\rho}}>\frac{h-3\rho}{h-2\rho}.$

The cs-holonomy map

$H^{cs}_{x_{0},y}:\overline{W^{u}_{f}(x_{0},\delta)}\to
W^{u}_{f}(y,c_{2})\subset f^{k+N_{3}}(W_{f}^{u}(x,\delta))$

is well defined by the choice of $N_{3}$ and $c_{2}$. Let $S$ be an
$(n_{\rho},\epsilon)$ u-separated subset of
$\overline{W^{u}_{f}(x_{0},\delta)}$ of maximal cardinality. Denote
$S^{\prime}=H^{cs}_{x_{0},y}(S)$. Since $W^{u}$ and $W^{cs}$ are uniformly
transverse foliations, by shrinking $\delta_{0}>0$ if necessary, there exists
some constant $c_{0}>1$ such that

$d_{f}^{cs}(x_{i},y_{i})\leq c_{0}\delta$

for any $x_{i}\in S$ and $y_{i}=H^{cs}_{x_{0},y}(x_{i})\in S^{\prime}$.

Since $S$ is $(n_{\rho},\epsilon)$ u-separated, we know that

$\\{W_{f}^{u}(f^{n_{\rho}}x_{i},\epsilon/2):x_{i}\in S\\}$

are disjoint balls contained in $f^{n_{\rho}}W^{u}_{f}(x_{0},\delta)$. Recall
that by Lyapunov stability of $f$, $0<\delta\ll\delta_{0}$ is chosen such that

$d^{cs}(f^{n_{\rho}}x_{i},f^{n_{\rho}}y_{i})<\delta_{0}/2.$

Then by the definition of $c_{3}=c_{3}(\epsilon,\delta)$ we have

$H^{cs}_{f^{n_{\rho}}x_{i},f^{n_{\rho}}y_{i}}(W_{f}^{u}(f^{n_{\rho}}x_{i},\epsilon/2))\supset
W^{u}(f^{n_{\rho}}y_{i},c_{3}).$

Moreover, $\\{W_{f}^{u}(f^{n_{\rho}}y_{i},c_{3}):y_{i}\in S^{\prime}\\}$ are
disjoint sets contained in $f^{n_{\rho}}W^{u}_{f}(y,c_{2})$.

By the choice of $N_{3}$,

$f^{N_{3}}(W^{u}_{f}(f^{n_{\rho}}y_{i},c_{3})\supset
W^{u}_{f}(f^{N_{3}+n_{\rho}}y_{i},2\delta).$

It follows that

$\\{W^{u}_{f}(f^{N_{3}+n_{\rho}}y_{i},2\delta):y_{i}\in S^{\prime}\\}$

are disjoint sets contained in $f^{N_{3}+n_{\rho}}W^{u}_{f}(y,c_{2})\subset
f^{n_{\rho}+k+2N_{3}}W^{u}_{f}(x,\delta)$. Since
$\\#S^{\prime}=\\#S>e^{n_{\rho}(h-2\rho)}$, we know that
$f^{n_{\rho}+k+2N_{3}}W^{u}_{f}(x,\delta)$ contains at least
$e^{n_{\rho}(h-2\rho)}$ disjoint sets of the form
$\\{W^{u}_{f}(z_{i},2\delta):z_{i}=f^{N_{3}+n_{\rho}}y_{i}\in M\\}.$ Then it
is easy to see that $f^{N(\rho,\delta)}W^{u}_{f}(x,\delta)$ where
$N(\rho,\delta)=n_{\rho}+N_{2}+2N_{3}$ contains at least
$e^{n_{\rho}(h-2\rho)}>e^{N(\rho,\delta)(h-3\rho)}$ disjoint sets of the form

$\\{W^{u}_{f}(w_{i},2\delta):w_{i}\in M\\}.$

Thus $f$ has uniform exponential growth along unstable foliation. ∎

###### Proof of Theorem B.

Theorem B is an immediate consequence of Proposition 4.2 and Theorem A. ∎

###  4.2. Examples

####  4.2.1. Time-one maps of frame flows

Let $M$ be a closed oriented $n$-dimensional manifold of negative sectional
curvature. Consider the unit tangent bundle $SM=\\{(x,v):x\in M,v\in
T_{x}M,\|v\|=1\\}$, and the frame bundle

$\displaystyle FM=\\{$ $\displaystyle(x,v_{0},v_{1},\cdots,v_{n-1}):x\in
M,v_{i}\in T_{x}M,$ $\displaystyle\\{v_{0},\cdots,v_{n-1}\\}\text{\ is a
positively oriented orthonormal frame at\ }x\\}.$

The geodesic flow $g^{t}:SM\to SM$ is defined as

$g^{t}(x,v)=(\gamma_{(x,v)}(t),\dot{\gamma}_{(x,v)}(t))$

where $\gamma_{(x,v)}(t)$ is the unique geodesic determined by the vector
$(x,v)$. The frame flow $f^{t}:FM\to FM$ is defined by

$f^{t}(x,v_{0},v_{1},\cdots,v_{n-1})=(g^{t}(x,v_{0}),\Gamma_{\gamma}^{t}(v_{1}),\cdots,\Gamma_{\gamma}^{t}(v_{n-1}))$

where $\Gamma_{\gamma}^{t}$ is the parallel transport along the geodesic
$\gamma_{(x,v_{0})}(t)$. There is a natural fiber bundle $\pi:FM\to SM$ that
takes a frame to its first vector, and then each fiber can be identified with
$SO(n-1)$. We have $\pi\circ f^{t}=g^{t}\circ\pi$ and $f^{t}$ acts
isometrically along the fibers.

Let $f=f^{1}$ be the time-one map of the frame flow $f^{t}$. Then $f$ is a PHD
with splitting $TFM=E_{f}^{s}\oplus E_{f}^{c}\oplus E_{f}^{u}$, where $\dim
E_{f}^{c}=1+\dim SO(n-1)$ and $f$ acts isometrically on the center bundle. If
in addition $f$ is topologically transitive, then Theorem B applies to $f$.
The following are the cases when $f$ is known to be ergodic with respect to
the normalized volume measure on $FM$, and hence topologically transitive.

######  Proposition 4.3.

(Cf. Theorem 0.2 in [7]) Let $f^{t}$ be the frame flow on an $n$-dimensional
compact smooth Riemannian manifold with sectional curvature between
$-\Lambda^{2}$ and $-\lambda^{2}$ for some $\Lambda,\lambda>0$. Then in each
of the following cases the flow and its time-one map are ergodic:

  * •

if the curvature is constant,

  * •

for a set of metrics of negative curvature which is open and dense in the
$C^{3}$ topology,

  * •

if $n$ is odd and $n\neq 7$,

  * •

if $n$ is even, $n\neq 8$, and $\lambda/\Lambda>0.93$,

  * •

if $n=7$ or $8$ and $\lambda/\Lambda>0.99023...$.

####  4.2.2. Semisimple translations on homogeneous spaces

In this example, we consider a special class of left translations on
homogenous spaces. Let $G$ be a connected semisimple Lie group, $\Gamma$ an
irreducible lattice in $G$, and $g\in G$. We say that $g$ is semisimple, if
$\mathrm{Ad}g|_{\mathfrak{h}^{0}_{\mathbb{C}}}$ is diagonalizable over
$\mathbb{C}$, where
$\mathfrak{h}^{0}_{\mathbb{C}}:=\bigoplus_{|\lambda|=1}E_{\lambda}$ (see (12)
in Section 5.2). If $\mathrm{Ad}g$ is non-quasiunipotent, then $T_{g}$ is a
$C^{\infty}$ PHD with bounded expansion in center direction. In fact, $T_{g}$
acts isometrically in the center direction. Moreover, $T_{g}$ is ergodic (in
fact, Kolmogorov by [12, 13] and Bernoulli by [25]) with respect to the Haar
measure on $G/\Gamma$, hence it is topologically transitive. Thus Theorem B
applies to $T_{g}$.

####  4.2.3. Time-one maps of geodesic flows on symmetric spaces of noncompact
type

If the symmetric space of noncompact type has rank one, then the geodesic flow
is an Anosov flow. In this case, the topological entropy is continuous at the
time-one maps of geodesic flows (and their perturbations) in the $C^{1}$
topology ([22, 21, 46]).

Now we consider symmetric spaces of noncompact type with rank $\geq 2$. Let
$(X,g)$ be a simply connected Riemannian symmetric space, then $X=G/K$, where
$G$ is the connected component of the isometry group of $X$ which acts
transitively on $X$, $K$ is the isotropy group of any point which is compact.
$X$ is called a symmetric space of noncompact type if $G$ is semisimple with
no compact factors and $K$ is a maximal compact subgroup of $G$. Assume
$\text{rank}(X)\geq 2$. Let $\Gamma\subset G$ be a cocompact lattice, then
$M=\Gamma\setminus X$ is a compact Riemannian manifold. Consider the geodesic
flow $g^{t}:SM\to SM$ on the unit tangent bundle of $M$ and its time-one map
$g=g^{1}$. Then $g$ is a $C^{\infty}$ PHD with higher dimensional center. We
note that the center direction has polynomial orbit growth and $g^{t}$ is not
ergodic with respect to the Liouville measure $m$ on $SM$.

For a fixed $p\in X$, choose a Weyl chamber $W\subset S_{p}X$. Then for each
$v$ in the interior of $W$ the orbit $Gv$ constitutes a closed set invariant
under the geodesic flow. The geodesic flow restricted to the submanifold
$SM_{v}:=\Gamma\setminus Gv\subset SM$ is mixing with respect to the
probability measure $m^{v}$ induced by the Liouville measure $m$ on $SM$.
Moreover, there exists a unique $v_{\text{max}}$ such that
$h_{\mathop{\hbox{\footnotesize\rm
top}}}(g^{t})=h_{\mathop{\hbox{\footnotesize\rm
top}}}(g^{t},SM_{v_{\text{max}}})$ and $g$ restricted to $SM_{v_{\text{max}}}$
is a $C^{\infty}$ PHD. See [29] for more details. At any point in
$SM_{v_{\text{max}}}$, the unstable manifolds of $g|SM_{v_{\text{max}}}$
coincides with the unstable manifolds of $g$. However, $g|SM_{v_{\text{max}}}$
is non-expanding (and hence of bounded expansion) in the center direction now,
i.e.,

$d^{cs}(g^{n}(x),g^{n}(y))\leq d^{cs}(x,y)$

for any $x,y$ on the same local center-unstable leaf (see Page 181 in [29]).
On the other hand, we know that $g^{t}$ is topologically transitive on
$SM_{v_{\text{max}}}$ as it is ergodic with respect to $m^{v_{\text{max}}}$.
One can check that Lemma 4.1 holds for a real number $0\leq k\leq n$ (see the
proof of Proposition 4.2). Furthermore, the proof of Proposition 4.2 goes
through with real $k$. Thus the conclusion of Theorem B applies to $g$.

##  5\. PHDs with subexponetial growth in the center direction

In this section, we consider various PHDs with subexponential growth in the
center direction. Recall that a $C^{1}$ PHD $f:M\to M$ has subexponential
growth in the center direction if for any $\epsilon>0$, there exists a
constant $C_{\epsilon}>0$ such that $\|D_{x}f^{n}|_{E_{f}^{c}(x)}\|\leq
C_{\epsilon}e^{n\epsilon}$ for any $n\in\mathbb{N}$ and any $x\in M$. The
following lemma is immediate.

######  Lemma 5.1.

If $f:M\to M$ is a $C^{1}$ PHD with subexponential growth in the center
direction, then for any $x\in M$, $\delta,\epsilon>0$ and $n\in{\mathbb{N}}$,

$f^{n}(W_{f}^{cs}(x,C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2))\subset
W^{cs}_{f}(f^{n}x,\delta).$

###  5.1. PHDs with property (EM)

In the first subsection, we prove Theorem C. Let $f:M\to M$ be a $C^{1}$
dynamically coherent partially hyperbolic diffeomorphism which has
subexponential growth in center direction and satisfies property (EM).

Given any $x_{0}\in M,0<\delta<\delta_{0}$ and $\epsilon>0$,
$R(x_{0},n,\epsilon,\delta)$ is called an _$(n,\epsilon,\delta)$ -rectangle
about $x_{0}$_ if it is in the form

$R(x_{0},n,\epsilon,\delta):=\bigcup_{y\in
W_{f}^{cs}(x_{0},C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2)}W^{u}_{f}(y,2\delta).$

As $W^{u}$ and $W^{cs}$ are uniformly transverse continuous foliations, there
exists $c_{0}=c_{0}(\delta_{0})>0$ such that for any $0<\delta<\delta_{0}$,

(6) $B(x_{0},c_{0}^{-1}C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2)\subset
R(x_{0},n,\epsilon,\delta)\subset\bigcup_{y\in
W^{u}_{f}(x_{0},3\delta)}W_{f}^{cs}(y,c_{0}C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2).$

Recall that for a compact Riemannian manifold $M$, $\phi\in
C^{\infty}(M,{\mathbb{R}})$ and $l\in{\mathbb{N}}$, the Sobolev norm is
defined as

(7) $\|\phi\|_{l,2}=\left(\sum_{0\leq j\leq
l}\int_{M}|\nabla^{j}\phi|^{2}d\mu\right)^{1/2},$

where $\nabla$ denotes the covariant derivative and $\mu$ denotes the volume
measure associated to $M$ (cf. [18, Definition 2.1]).

The following is lemma is useful. See [24, Lemma 3.3] for proof.

######  Lemma 5.2.

Let $M$ be a $d$-dimensional compact smooth Riemannian manifold and
$l\in{\mathbb{N}}$. Then there exist $r_{0}>0$ and $M_{l,d}>0$ such that the
following holds: for any $x\in M$ and $0<\epsilon,r<r_{0}$, there exists
$\phi_{\epsilon}\in C^{\infty}(M,[0,1])$ such that

  1. (1)

$\phi_{\epsilon}\equiv 1$ on $B(x,r)$;

  2. (2)

$\phi_{\epsilon}\equiv 0$ on $B(x,r+\epsilon)^{c}$;

  3. (3)

$\|\phi_{\epsilon}\|_{l}\leq M_{l,d}\epsilon^{-d-l}$;

  4. (4)

$\|\phi_{\epsilon}\|_{l,2}\leq M_{l,d}\epsilon^{-d-l}$.

The following is a crucial lemma on effective density of the unstable
foliation.

######  Lemma 5.3.

(Effective density) For any $0<\delta<\delta_{0}$ and $\epsilon>0$, there
exist $N_{1}\in{\mathbb{N}}$ and $D>0$ such that for any $n\geq N_{1}$, $k\geq
Dn\epsilon$ and any $x_{0},x\in M$, $f^{k}W^{u}_{f}(x,\delta)\cap
R(x_{0},n,\epsilon,\delta)$ has a connected component of the form
$W^{u}_{f}(y,2\delta)$ for some $y\in
W_{f}^{cs}(x_{0},C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2)$.

###### Proof.

Fix arbitrary $0<\delta<\delta_{0}$ and $\epsilon>0$. At first, we show that
there exists $N_{1}^{\prime}\in{\mathbb{N}}$ and $D>0$ such that for any
$n\geq N_{1}^{\prime}$, $k\geq Dn\epsilon$ and any $x,x_{0}\in M$,

(8) $f^{k}W^{u}_{f}(x,\delta/2)\cap
B(x_{0},c_{0}^{-1}C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2)\neq\emptyset$

where $c_{0}$ satisfies (6).

Let
$A=W^{u}_{f}(x,\delta/2),B=B(x_{0},c_{0}^{-1}C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2)$,
$\chi_{A}$ and $\chi_{B}$ their characteristic functions. Applying Lemma 5.2
for $B(x,r)=A$ (resp. $B(x,r)=B$), $\epsilon=e^{-\lambda^{\prime}k}$ for some
sufficiently small $\lambda^{\prime}$ specified later, we obtain $C^{\infty}$
functions $\psi_{k}$ (resp. $\phi_{k}$).

Applying property (EM) (1) to $C^{\infty}$ functions $\phi_{k}$ and
$\psi_{k}$, we have

(9)
$\displaystyle\int_{W_{f}^{u}(x,\delta)}\phi_{k}(f^{k}(p))\psi_{k}(p)dm_{x}^{u}(p)\geq$
$\displaystyle\int\phi_{k}dm\int_{W_{f}^{u}(x,\delta)}\psi_{k}dm_{x}^{u}-Ce^{-k\alpha}M_{l,d}M_{l,u}e^{\lambda^{\prime}k(2l+d+u)}$

for any $k\geq 0$ where $u=\dim W_{f}^{u}(x,\delta)$. By choosing
$\lambda^{\prime}$ small enough such that
$\alpha-(2l+d+u)\lambda^{\prime}>\lambda^{\prime}$, we see from (9) that

$\displaystyle\int_{W_{f}^{u}(x,\delta)}\phi_{k}(f^{k}(p))\psi_{k}(p)dm_{x}^{u}(p)\geq\int\phi_{k}dm\int_{W_{f}^{u}(x,\delta)}\psi_{k}dm_{x}^{u}-C_{1}e^{-\lambda^{\prime}k}.$

Then we have

$\displaystyle\int_{W_{f}^{u}(x,\delta)}\chi_{B}(f^{k}(p))\chi_{A}(p)dm_{x}^{u}(p)$
$\displaystyle\geq$
$\displaystyle\int_{W_{f}^{u}(x,\delta)}\phi_{k}(f^{k}(p))\psi_{k}(p)dm_{x}^{u}(p)-\int|\psi_{k}-\chi_{A}|dm_{x}^{u}-\int|\phi_{k}-\chi_{B}|dm$
$\displaystyle\geq$
$\displaystyle\int\phi_{k}dm\int_{W_{f}^{u}(x,\delta)}\psi_{k}dm_{x}^{u}-C_{1}e^{-\lambda^{\prime}k}-C_{2}e^{-\lambda^{\prime}k}$
$\displaystyle\geq$ $\displaystyle
C_{3}(\delta/2)^{u}\cdot(c_{0}^{-1}C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2)^{d}-C_{4}e^{-\lambda^{\prime}k}$
$\displaystyle=$ $\displaystyle
C_{5}e^{-nd\epsilon}-C_{4}e^{-\lambda^{\prime}k}>0$

if $n\geq N_{1}^{\prime}$, $k\geq Dn\epsilon$ for some sufficiently large
$N_{1}^{\prime}\in{\mathbb{N}}$ and some $D>\frac{d}{\lambda^{\prime}}$. Then
(8) follows immediately.

Now set $N_{1}=\max\\{N_{1}^{\prime},N_{1}^{\prime\prime}\\}$, where
$N_{1}^{\prime\prime}$ satisfies that, for any $k\geq
DN_{1}^{\prime\prime}\epsilon$,

$f^{-k}W_{f}^{u}(w,4\delta)\subset W^{u}_{f}(f^{-k}w,\delta/2)$

for any $w\in M$. Take $n\geq N_{1}$ and $k\geq Dn\epsilon$. In view of (6)
and (8), there exists

(10) $\displaystyle z\in f^{k}W^{u}_{f}(x,\delta/2)\cap
B(x_{0},c_{0}^{-1}C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2)\subset
f^{k}W^{u}_{f}(x,\delta/2)\cap R(x_{0},n,\epsilon,\delta).$

Therefore, there exists $y\in
W_{f}^{cs}(x_{0},C_{\epsilon}^{-1}e^{-n\epsilon}\delta/2)$ such that $z\in
W^{u}_{f}(y,2\delta)$. Take any $w\in W^{u}_{f}(y,2\delta)$, and it is clear
that $d^{u}(z,w)\leq 4\delta.$ Then by the choice of $N_{2}^{\prime\prime}$,
$d^{u}(f^{-k}w,f^{-k}z)\leq\delta/2$. In view of (10), we have

$f^{-k}w\in W^{u}_{f}(x,\delta).$

This implies $W^{u}_{f}(y,2\delta)\subset f^{k}W^{u}_{f}(x,\delta)\cap
R(x_{0},n,\epsilon,\delta)$ and completes the proof of the lemma. ∎

######  Proposition 5.4.

Let $f:M\to M$ be a $C^{1}$ dynamically coherent PHD which has subexponential
growth in the center direction and satisfies property (EM). Then $f$ has
uniform exponential growth along unstable foliation.

###### Proof.

Denote by $h$ the unstable topological entropy of $f$, i.e.,
$h=h^{u}_{{\mathop{\hbox{\footnotesize\rm top}}}}(f)$. Since $f$ has
subexponential growth in the center direction, Proposition 2.6 tells us that
$h=h_{{\mathop{\hbox{\footnotesize\rm top}}}}(f)$, the topological entropy of
$f$. Then by the definition of unstable topological entropy and Remark 2.2,
for any $\rho>0$, we can find $x_{0}\in M$ and $0<\delta<\delta_{0}$ small
enough such that

$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}(f,\overline{W^{u}_{f}(x_{0},\delta)})>h-\rho.$

Pick $0<\epsilon\ll\delta$ and $N_{2}\in{\mathbb{N}}$ large enough such that
if $n\geq N_{2}$, then

$N^{u}(f,n,\epsilon,x_{0},\delta)>e^{n(h-2\rho)},$

where $N^{u}(f,n,\epsilon,x_{0},\delta)$ denotes the cardinality of a maximal
$(n,\epsilon)$ u-separated subset $S$ of $\overline{W^{u}_{f}(x_{0},\delta)}$,
i.e., $\\#S=N^{u}(f,n,\epsilon,x_{0},\delta)>e^{n(h-2\rho)}$ where $\\#S$
denotes the cardinality of $S$.

We note that by dynamical coherence of $f$, the center-stable foliation
$W_{f}^{cs}$ exists and it is continuous. Shrinking $\delta_{0}$ if necessary,
for any $y\in W^{cs}_{f}(x,c_{0}\delta_{0})$, the $cs$-holonomy map
$H_{x,y}^{cs}$ is well defined such that

$H^{cs}_{x,y}(\overline{W^{u}_{f}(x,\delta)})\subset
W^{u}_{f}(y,c_{2}(\delta))$

and

$H^{cs}_{x,y}(W^{u}_{f}(x,\epsilon/2))\supset
W^{u}_{f}(y,c_{3}(\epsilon,\delta))$

for some constants $c_{2}=c_{2}(\delta)>2\delta$ and
$c_{3}=c_{3}(\epsilon,\delta)>0$. Then take $N_{3}\in{\mathbb{N}}$ large
enough such that for any $n\geq N_{3}$ and for any $x\in M$:

  1. (1)

$f^{n}(W^{u}_{f}(x,c_{3}))\supset W^{u}_{f}(f^{n}(x),2\delta)$;

  2. (2)

$f^{n}(W^{u}_{f}(x,\delta))\supset W^{u}_{f}(f^{n}(x),2c_{2})$.

Now take any $x\in M$ and $\bar{\epsilon}$ small enough to be specified later.
Applying Lemma 5.3 with $x=f^{N_{3}}x$, $x_{0}=x_{0}$, $\delta=\delta$ and
$\epsilon=\bar{\epsilon}$, there exists $N_{1}\in{\mathbb{N}}$ depending only
on $\delta$ and $\bar{\epsilon}$ such that for any $n\geq N_{1}$

(11) $f^{k}(W_{f}^{u}(f^{N_{3}}x,\delta))\cap
W_{f}^{cs}(x_{0},C_{\bar{\epsilon}}^{-1}e^{-n\bar{\epsilon}}\delta/2)\neq\emptyset$

where $k=Dn\bar{\epsilon}$.

Finally, we take $\bar{\epsilon}$ small enough and $n_{\rho}\in{\mathbb{N}}$
large enough such that $n_{\rho}>\max\\{N_{1},N_{2},N_{3}\\}$ and

$\frac{n_{\rho}}{2N_{3}+Dn_{\rho}\bar{\epsilon}+n_{\rho}}>\frac{h-3\rho}{h-2\rho}.$

The remaining part of the proof of the proposition is divided into three
steps:

Step $1$. Denote $k=Dn_{\rho}\bar{\epsilon}$. It follows from (11) that there
exists some point

$y\in f^{k+N_{3}}(W_{f}^{u}(x,\delta))\cap
W_{f}^{cs}(x_{0},C_{\bar{\epsilon}}^{-1}e^{-n\bar{\epsilon}}\delta/2).$

By the choice of $N_{3}$ and the fact $c_{2}>2\delta$, we have

$f^{k+N_{3}}(W_{f}^{u}(x,\delta))\supset W^{u}_{f}(y,c_{2}).$

Hence the cs-holonomy map

$H^{cs}_{x_{0},y}:\overline{W^{u}_{f}(x_{0},\delta)}\to
W^{u}_{f}(y,c_{2})\subset f^{k+N_{3}}(W_{f}^{u}(x,\delta))$

is well defined by the choice of $c_{2}$.

Step $2$. Let $S$ be an $(n_{\rho},\epsilon)$ u-separated subset of
$\overline{W^{u}_{f}(x_{0},\delta)}$ of maximal cardinality. Denote
$S^{\prime}=H^{cs}_{x_{0},y}(S)$. By (6), we have

$d_{f}^{cs}(x_{i},y_{i})\leq
c_{0}C_{\bar{\epsilon}}^{-1}e^{-n\bar{\epsilon}}\delta/2$

for any $x_{i}\in S$ and $y_{i}=H^{cs}_{x_{0},y}(x_{i})\in S^{\prime}$.

Since $S$ is $(n_{\rho},\epsilon)$ u-separated, we know that

$\\{W_{f}^{u}(f^{n_{\rho}}x_{i},\epsilon/2):x_{i}\in S\\}$

are disjoint balls contained in
$f^{n_{\rho}}\overline{W^{u}_{f}(x_{0},\delta)}$. By Lemma 5.1,

$d^{cs}(f^{n_{\rho}}x_{i},f^{n_{\rho}}y_{i})<c_{0}\delta.$

Then we have

$H^{cs}_{f^{n_{\rho}}x_{i},f^{n_{\rho}}y_{i}}(W_{f}^{u}(f^{n_{\rho}}x_{i},\epsilon/2))\supset
W^{u}(f^{n_{\rho}}y_{i},c_{3})$

by the definition of $c_{3}$. Moreover,

$\\{W_{f}^{u}(f^{n_{\rho}}y_{i},c_{3}):y_{i}\in S^{\prime}\\}$

are disjoint sets contained in $f^{n_{\rho}}W^{u}_{f}(y,c_{2})$.

Step $3$. By the choice of $N_{3}$,

$f^{N_{3}}(W^{u}_{f}(f^{n_{\rho}}y_{i},c_{3})\supset
W^{u}_{f}(f^{N_{3}+n_{\rho}}y_{i},2\delta).$

It follows that

$\\{W^{u}_{f}(f^{N_{3}+n_{\rho}}y_{i},2\delta):y_{i}\in S^{\prime}\\}$

are disjoint sets contained in $f^{N_{3}+n_{\rho}}W^{u}_{f}(y,c_{2})\subset
f^{n_{\rho}+Dn_{\rho}\bar{\epsilon}+2N_{3}}W^{u}_{f}(x,\delta)$. Recall that
$\bar{\epsilon}$ is depending on $\rho$. Set

$N(\rho,\delta)=n_{\rho}+Dn_{\rho}\bar{\epsilon}+2N_{3}.$

Since $\\#S^{\prime}=\\#S>e^{n_{\rho}(h-2\rho)}>e^{N(\rho,\delta)(h-3\rho)}$,
we know $f^{N(\rho,\delta)}W^{u}_{f}(x,\delta)$ contains at least
$e^{N(\rho,\delta)(h-3\rho)}$ disjoint sets

$\\{W^{u}_{f}(f^{N_{3}+n_{\rho}}y_{i},2\delta):y_{i}\in S^{\prime}\\}$

This proves the proposition by setting $z_{i}=f^{N_{3}+n_{\rho}}y_{i}$. ∎

###### Proof of Theorem C.

Theorem C is an immediate consequence of Proposition 5.4 and Theorem A. ∎

###  5.2. Homogeneous dynamics

####  5.2.1. Definitions and basic properties

Let $G$ be a connected semisimple Lie group, and $g\in G$. For
$\lambda\in\mathbb{C}$, set

$E_{\lambda}:=\\{X\in\mathfrak{g}_{\mathbb{C}}:(\mathrm{Ad}(g)-\lambda
I)^{j}X=0\text{\ for some\ }j\in\mathbb{N}-\\{0\\}\\},$

where $\mathfrak{g}_{\mathbb{C}}$ denotes the complexification of
$\mathfrak{g}$. Let $\mathfrak{h}^{+},\mathfrak{h}^{-}$ be the subalgebras of
$\mathfrak{g}$ with complexifications

(12)
$\mathfrak{h}^{+}_{\mathbb{C}}:=\bigoplus_{|\lambda|>1}E_{\lambda},\quad\mathfrak{h}^{-}_{\mathbb{C}}:=\bigoplus_{|\lambda|\leq
1}E_{\lambda},$

and let $H^{+},H^{-}$ be the corresponding subgroups of $G$. Then $H^{+}$ is
the unstable horospherical subgroup for $g$, that is,

$H^{+}=\\{h\in G:\lim_{k\to+\infty}g^{-k}hg^{k}=e\\}$

where $e\in G$ is the identity of $G$. $H^{+}$ is a closed, connected subgroup
of $G$. Then $\mathrm{Ad}g$ (or just $g$ for short) is called _non-
quasiunipotent_ if $H^{+}$ is nontrivial. Let $d^{+}$(resp. $d^{-}$) be the
right invariant metric on $H^{+}$(resp. $H^{-}$) induced from the norm on
$\mathfrak{h}^{+}$(resp. $\mathfrak{h}^{-}$) inherited from that of
$\mathfrak{g}$. Let $B_{r}^{+}$(resp. $B_{r}^{-}$) denote the corresponding
metric ball in $(H^{+},d^{+})$ (resp. $(H^{-},d^{-})$) centered at the
identity $e\in G$ with radius $r$.

Let $\Gamma$ be a cocompact lattice in $G$, then we have a compact homogenous
space $G/\Gamma$. By compactness, there exist $\delta_{0}>0$ and $c_{0}>1$
such that for any $0<\delta<\delta_{0}$ and any $x\in G/\Gamma$, the map

$\psi_{x}:B_{\delta}^{+}\times B_{\delta}^{-}\rightarrow
G/\Gamma,\quad\psi_{x}(h^{+},h^{-})=h^{+}h^{-}x,$

is a diffeomorphism onto its image, and moreover

$B(x,c_{0}^{-1}\delta)\subset B_{\delta}^{+}B_{\delta}^{-}x\subset
B(x,c_{0}\delta).$

Denote the conjugate action of $g$ on $G$ as $C(g)$. Then $C(g)$ defines an
expanding automorphism on $H^{+}$, and an at most polynomially expanding
automorphism of $H^{-}$. Below is an explicit description of the action of
$C(g)$ on $H^{+}$ and $H^{-}$.

######  Lemma 5.5.

  1. (1)

There exist $\lambda\in(0,1)$ and $N_{1}\in{\mathbb{N}}$ such that
$C(g^{-n})h\in B_{\lambda^{n}\delta}^{+}$ whenever $n\geq
N_{1},0<\delta\leq\delta_{0}$ and $h\in B_{\delta}^{+}$.

  2. (2)

There exist $c_{1},\kappa>0$ such that
$C(g^{i})(B_{c_{1}n^{-\kappa}\delta}^{-})\subset B_{\delta}^{-}$ when
$0<\delta\leq\delta_{0}$ and $1\leq i\leq n$.

###### Proof.

All these properties are easily checked by passing to the Lie algebras
$\mathfrak{h}^{+},\mathfrak{h}^{-}$ using the exponential maps. ∎

Let $T_{g}:G/\Gamma\to G/\Gamma$ be the dynamical system induced by the left
translation of $T_{g}(x)=gx$, for any $x\in G/\Gamma$. Assume that
$\mathrm{Ad}g$ is nonquasiunipotent. Then $T_{g}$ is a $C^{\infty}$ partially
hyperbolic diffeomorphism. The local unstable and center-stable manifolds of
$T_{g}$ at $x\in G/\Gamma$ of size $\delta>0$ are given by

$W^{u}_{g}(x,\delta)=B_{\delta}^{+}x,\quad
W^{cs}_{g}(x,\delta)=B_{\delta}^{-}x.$

In the following, we will use both notations $W^{u}_{g}(x,\delta)$ and
$B_{\delta}^{+}x$ (resp. $W^{cs}_{g}(x,\delta)$ and $B_{\delta}^{-}x$)
interchangeably for the local unstable (resp. local center-stable) manifold of
$T_{g}$ at $x\in G/\Gamma$. $T_{g}$ has polynomial (hence subexponential)
growth in the center direction, by Lemma 5.5 and the discussion before it.

The topological entropy of $T_{g}$ can be calculated by the formula:

$h_{\mathop{\hbox{\footnotesize\rm
top}}}(g)=h_{\mu_{G}}(g)=\log|\det(\mathrm{Ad}g|_{\mathfrak{h}^{+}})|=\sum_{|\lambda|>1}\log|\lambda|$

where the sum is taken over all the eigenvalues of
$\mathrm{Ad}(g)|_{\mathfrak{h}^{+}}$ counted with multiplicity, and $\mu_{G}$
is the normalized Haar measure on $G/\Gamma$. In fact, the above formula holds
in a more general setting when $\Gamma$ is any lattice of $G$ (cf. [4] or
Corollary 2.5.7 in [35]).

####  5.2.2. Exponential mixing

Now we recall the exponential mixing property of $T_{g}$. To begin, let us
recall that for a Riemannian manifold $M$, $f\in
C_{c}^{\infty}(M,{\mathbb{R}})$ and $l\in{\mathbb{N}}$, the Sobolev norm is
defined in (7).

######  Proposition 5.6.

(Exponential mixing) Let $G$ be a connected semisimple Lie group without
compact factors, $\Gamma$ be an irreducible cocompact lattice of $G$ and $g\in
G$ be non-quasiunipotent. Then there exist $\mu,E,l,\theta>0$ such that for
any $x\in G/\Gamma$, $\psi\in C^{\infty}(G/\Gamma,{\mathbb{R}})$ and $f\in
C^{\infty}_{c}(H^{+},{\mathbb{R}})$ such that the map $\pi_{x}:g\rightarrow
gx$ is injective on $\text{supp}(f)$, and for any $n\in{\mathbb{N}}$ one has

$\left|\int_{H^{+}}f(h)\psi(g^{n}hx)d\nu_{H^{+}}(h)-\int_{H^{+}}fd\nu_{H^{+}}\int_{G/\Gamma}\psi
d\mu_{G}\right|\leq C(f,\psi)e^{-\mu n},$

where $\nu_{H^{+}}$ is a bi-invariant Haar measure on $H^{+}$, $\mu_{G}$ is
the normalized Haar measure on $G/\Gamma$, and

$C(f,\psi)=E\|f\|_{l,2}\|\psi\|_{l,2}\left(\max_{x\in
G/\Gamma}\|\nabla\psi(x)\|\int_{H^{+}}|f|d\nu_{H^{+}}\right)^{\theta}.$

In [28, Proposition 2.4.8], the authors obtained the same result for
$(G/\Gamma,g^{t})$ where $g^{t}$ is a one-parameter subgroup of $G$, which is
derived from the Condition (EM) [28, Section 2.4.1]. In our discrete setting,
the authors in [15, Propositions 3.1] provide a proof for a spectral gap
result for the regular representation of $G$ on the space
$L_{0}^{2}(G/\Gamma)$, from which the Condition (EM) follows.

We point out that the coefficient $C(f,\psi)$ is slightly different from that
in (1). Nevertheless, Lemma 5.3 can be proved analogously. See also the proof
of [24, Proposition 3.2], [5, Proposition 3.3] and [15, Lemma 3.5] where
indeed $C(f,\psi)$ is used.

Thus Propositions 5.4 and 5.6 also hold. As a corollary, we have:

######  Proposition 5.7.

Let $G$ be a connected semisimple Lie group without compact factors, $\Gamma$
be an irreducible cocompact lattice of $G$ and $g\in G$ be non-quasiunipotent.
Then $(G/\Gamma,g)$ has uniform exponential growth along unstable foliation.

Proposition 5.7 can be extended to a more general setting.

######  Proposition 5.8.

Let $G$ be a connected semisimple Lie group, $\Gamma$ be a cocompact lattice
of $G$, and $g\in G$ be non-quasiunipotent. Then $(G/\Gamma,g)$ has uniform
exponential growth along unstable foliation.

###### Proof.

Let $\widetilde{G}$ be the simply connected cover of $G$, $\widetilde{\Gamma}$
be the inverse image of $\Gamma$ in $\widetilde{G}$ and $\widetilde{g}$ be any
inverse of $g$ in $\widetilde{G}$. Then it is easily seen that there is an
isomorphism of systems

$(\widetilde{G}/\widetilde{\Gamma},\widetilde{g})\cong(G/\Gamma,g).$

Hence we may replace $G$ by its simply connected cover $\widetilde{G}$ and
$\Gamma$ by its inverse image in $\widetilde{G}$. Write

$G=G^{nc}\times K,\text{ with $K$ compact and $G^{nc}$ has no compact factors.
}$

Then $q(\Gamma)$ is a lattice in $G^{nc}$, where $q$ denotes the projection
from $G$ to $G^{nc}$. According to [41, Theorem 5.22], $G^{nc}$ can be
decomposed as

$G^{nc}=G_{1}\times\cdots\times G_{m}$

such that $\Gamma\cap G_{i}$ are irreducible lattices in $G_{i}(1\leq i\leq
m)$. Write $g=(g_{1},\ldots,g_{m},k)$ with $g_{i}\in G_{i}$ and $k\in K$. We
may assume that $g_{i}$ is non-quasiunipotent for $1\leq i\leq j$ and $g_{i}$
is quasiunipotent for $j<i\leq m$. Then set

$G^{\prime}=G_{1}\times\cdots\times G_{j},\quad
G^{\prime\prime}=G_{j+1}\cdots\times G_{m}\times K.$

By assumption $G^{\prime}$ is nontrivial and we write
$g=(g^{\prime},g^{\prime\prime})$ with $g^{\prime}\in G^{\prime}$ non-
quasiunipotent and $g^{\prime\prime}\in G^{\prime\prime}$ quasiunipotent.
Denote the projection of $G$ to $G^{\prime}$ as $p$. Using [41, Theorem 5.22]
again, we obtain that the group $p(\Gamma)$ is commensurable with
$\prod_{1\leq i\leq j}(\Gamma\cap G_{i})$, hence is a lattice in $G^{\prime}$.
Thus we know that $G/\Gamma$ is a fiber bundle over $G^{\prime}/p(\Gamma)$
with fiber $G^{\prime\prime}/(\Gamma\cap G^{\prime\prime})$.

Firstly, we study the system $(G^{\prime}/p(\Gamma),g^{\prime})$. By
definition, there is a finite index subgroup $\Gamma^{\prime}\subset
p(\Gamma)$ such that $\Gamma^{\prime}=\prod_{1\leq i\leq j}\Gamma_{i}$ with
$\Gamma_{i}$ irreducible lattices in $G_{i}$. By Proposition 5.7, each
$(G_{i}/\Gamma_{i},g_{i})$ has uniform exponential growth along unstable
foliation. Then $(G^{\prime}/\Gamma^{\prime},g^{\prime},d^{\prime})$ is a
product of systems $(G_{i}/\Gamma_{i},g_{i},d_{i})$ where it is more
convenient to work with the metrics

$d^{\prime}((x_{i}),(y_{i}))=\max_{1\leq i\leq j}d_{i}(x_{i},y_{i}).$

Note that $h_{{\mathop{\hbox{\footnotesize\rm top}}}}(g^{\prime})=\sum_{1\leq
i\leq j}h_{{\mathop{\hbox{\footnotesize\rm top}}}}(g_{i})$. It is not hard to
verify that $G^{\prime}/\Gamma^{\prime}$ has uniform exponential growth along
unstable foliation by choosing

$N(\rho,\delta)=\max_{1\leq i\leq j}N_{i}(\rho,\delta).$

Now taking into account that there exists a locally isometric finite covering
map $\varphi:G^{\prime}/\Gamma^{\prime}\to G^{\prime}/p(\Gamma)$, we see that
$G^{\prime}/p(\Gamma)$ has uniform exponential growth along unstable
foliation.

Now consider the fiber system $(G/\Gamma,g)$ over
$(G^{\prime}/p(\Gamma),g^{\prime})$ with fiber $G^{\prime\prime}/(\Gamma\cap
G^{\prime\prime})$. Note that $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}(g)=h_{{\mathop{\hbox{\footnotesize\rm top}}}}(g^{\prime})$ and the
Lyapunov exponents of $g$ along the direction of fibers are zero. Notice that
the unstable horospherical subgroup $H^{+}$ of $g$ is a subgroup of
$G^{\prime}$. If $x=h\Gamma=h^{\prime}h^{\prime\prime}\Gamma\in G/\Gamma$ with
$h^{\prime}\in G^{\prime}$ and $h^{\prime\prime}\in G^{\prime\prime}$, then
for $B^{+}_{\delta}\subset H^{+}$,

$p(B_{\delta}^{+}h\Gamma)=B_{\delta}^{+}p(h\Gamma)=B_{\delta}^{+}h^{\prime}p(\Gamma).$

So $p$ is in fact a local isometry between $W^{u}_{g}(x)\to
W^{u}_{g^{\prime}}(p(x))$. As $p\circ T_{g}=T_{g^{\prime}}\circ p$, it is
clear that $(G/\Gamma,g)$ has uniform exponential growth along unstable
foliation if $(G^{\prime}/p(\Gamma),g^{\prime})$ has. ∎

###### Proof of Theorem D.

If $g$ is quasiunipotent, the topological entropy of $T_{g}$ is $0$, then
there is nothing to prove.

Assume that $g$ is non-quasiunipotent. Then by Proposition 5.8, $(G/\Gamma,g)$
has uniform exponential growth along unstable foliation. Moreover, it has
subexponential growth in the center direction. Now Theorem D is an immediate
corollary of Theorem A(2). ∎

####  5.2.3. An improvement of Theorem D

######  Theorem D’.

Let $G$ be a connected semisimple Lie group, $\Gamma$ be a cocompact lattice
of $G$, and $g\in G$. Then the topological entropy function
$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(G/\Gamma)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at
$T_{g}$.

###### Proof.

Let $\epsilon>0$ and $f$ be $C^{1}$-close enough to $T_{g}$ which will be
specified later. By Ruelle inequality, for any $f$-ergodic measure $\nu$ and
any $N\in{\mathbb{N}}$,

(13) $\displaystyle h_{\nu}(f)$
$\displaystyle\leq\sum_{\lambda_{j}(\nu)>0}\lambda_{j}(\nu)m_{j}(\nu)$
$\displaystyle\leq\dim E_{f}^{c}\cdot\frac{1}{N}\max_{x\in
G/\Gamma}\log\|D_{x}f^{N}|_{E_{f}^{c}(x)}\|+\max_{x\in
G/\Gamma}\log\det(D_{x}f|_{E_{f}^{u}(x)}).$

Note that $\|D_{x}T_{g}^{N}|_{E_{g}^{c}(x)}\|$ is constant in $x\in G/\Gamma$.
Choose $N$ large enough such that
$\frac{1}{N}\log\|D_{x}T_{g}^{N}|_{E_{g}^{c}(x)}\|<\frac{\epsilon}{3\dim
E_{g}^{c}}$ for any $x\in G/\Gamma$. Now choose $f$ $C^{1}$-close enough to
$T_{g}$ such that for any $x\in G/\Gamma$,

(14)
$\Big{|}\log\|D_{x}f^{N}|_{E_{f}^{c}(x)}\|-\log\|D_{x}T_{g}^{N}|_{E_{g}^{c}(x)}\|\Big{|}<\frac{\epsilon}{3\dim
E_{f}^{c}}$

and

(15)
$\Big{|}\log\det(D_{x}f|_{E_{f}^{u}(x)})-\log\det(D_{x}T_{g}|_{E_{g}^{u}(x)})\Big{|}<\frac{\epsilon}{3}.$

Note that

(16) $h_{\mathop{\hbox{\footnotesize\rm
top}}}(T_{g})=\log\det(\mathrm{Ad}g|_{\mathfrak{h}^{+}})=\log\det(D_{x}T_{g}|_{E_{g}^{u}(x)})$

for any $x\in G/\Gamma$. Combining (13), (14), (15) and (16), we have
$h_{\nu}(f)<h_{\mathop{\hbox{\footnotesize\rm top}}}(T_{g})+\epsilon$. Then
$h_{\mathop{\hbox{\footnotesize\rm
top}}}(f)=\sup_{\nu\in{\mathcal{M}}_{f}^{e}(G/\Gamma)}h_{\nu}(f)\leq
h_{\mathop{\hbox{\footnotesize\rm top}}}(T_{g})+\epsilon$. This proves the
upper semicontinuity of $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(G/\Gamma)\to{\mathbb{R}}^{+}\cup\\{0\\}$ at $T_{g}$.

The lower semicontinuity of $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(G/\Gamma)\to{\mathbb{R}}^{+}\cup\\{0\\}$ at $T_{g}$ is obtained
in Theorem D. We give a simple proof when the system after perturbation is
$C^{r},r>1$. For any $\epsilon>0$, choose a $C^{r},r>1$ diffeomorphism
$f:G/\Gamma\to G/\Gamma$ which is $C^{1}$-close enough to $T_{g}$ such that
for any $x\in G/\Gamma$ (15) is satisfied. As $f$ is $C^{r},r>1$, let $\mu$ be
a Gibbs u-state for $f$, i.e., an $f$-invariant measure whose conditional
measures along unstable leaves are absolutely continuous with respect to the
Lebesgue measure on the unstable leaves. The existence of $\mu$ is guaranteed
by [39]. Then we have

(17) $\displaystyle h_{\mathop{\hbox{\footnotesize\rm top}}}(f)\geq
h_{\mathop{\hbox{\footnotesize\rm top}}}^{u}(f)\geq h^{u}_{\mu}(f)$
$\displaystyle=$
$\displaystyle\int_{G/\Gamma}\log\det(D_{x}f|_{E_{f}^{u}(x)})d\mu(x)\geq\log\det(DT_{g}|_{E_{g}^{u}})-\epsilon/3$
$\displaystyle=$ $\displaystyle h_{\mathop{\hbox{\footnotesize\rm
top}}}(T_{g})-\epsilon/3.$

This proves the lower semicontinuity of $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{r}(G/\Gamma)\to{\mathbb{R}}^{+}\cup\\{0\\}$ at $T_{g}$, $r>1$. ∎

######  Remark 5.9.

(16) is crucial for the above argument to give upper semicontinuity. In
general, PHDs considered in Theorem C do not necessarily satisfy (16). As far
as we know, a Gibbs u-state satisfying the entropy estimate in (17) exists
only for $C^{r},r>1$ PHDs.

####  5.2.4. Further questions

Though the most work in this paper is done on the unstable foliation of
$T_{g}$, we remark that the center distribution of $T_{g}$ corresponding to
the subalgebra
$\mathfrak{h}^{0}:=\bigoplus_{|\lambda|=1}E_{\lambda}\cap\mathfrak{g}$, is
$C^{\infty}$. Then the center foliation is $C^{\infty}$ and hence plaque
expansive (cf. [19]). Thus there exists a neighborhood $\mathcal{U}$ of
$T_{g}$ in the $C^{1}$ topology, such that for each $f\in\mathcal{U}$,

  1. (1)

there exists a homeomorphism $h_{f}:G/\Gamma\to G/\Gamma$, called a leaf
conjugacy, satisfying $h_{f}(W^{c}_{g}(x))=W^{c}_{f}(h_{f}(x))$ and
$h_{f}\circ T_{g}(W^{c}_{g}(x))=f\circ h_{f}(W^{c}_{g}(x))$. Moreover, the
leaf conjugacy $h_{f}$ is close to the identity map in the $C^{0}$ topology.

  2. (2)

$f$ is dynamically coherent, i.e., there exist $f$-invariant foliations
$W^{cs}_{f}$ and $W^{cu}_{f}$ tangent to $E^{c}_{f}\oplus E^{s}_{f}$ and
$E^{c}_{f}\oplus E^{u}_{f}$ respectively; by intersecting their leaves we then
obtain an $f$-invariant foliation $W^{c}_{f}$.

A further question arises about the continuity property of the topological
entropy at each $f\in\mathcal{U}$. Clearly, the properties of $f$ listed above
will be fundamental. However, other properties of $f$ such as property (EM)
should be developed, for which we have no explicit ideas now.

On the other hand, one can study a smooth time change of a partially
hyperbolic homogeneous flow. Let $(G/\Gamma,g_{t})$ be a partially hyperbolic
homogeneous flow, i.e., $g_{t}$ is a one-parameter subgroup of $G$ for which
each element is non-quasiunipotent. Let $\tau:G/\Gamma\to{\mathbb{R}}$ be a
smooth positive function. The time-changed flow is defined as
$g^{\tau}_{t}(x):=g_{\alpha(x,t)}(x)$, where $\alpha(x,t)$ is determined by

$\int_{0}^{\alpha(x,t)}\tau(g_{s}(x))ds=t.$

Then the time-one map $g^{\tau}_{1}$ is also a $C^{\infty}$ PHD whose center,
center-stable and center-unstable foliations all coincide with those of
$T_{g}$. However, we do not know if $g^{\tau}_{1}$ has property (EM).

###  5.3. Toral automorphisms

Suppose that a toral automorphism $T_{A}:\mathbb{T}^{d}\to\mathbb{T}^{d}$ is
induced by a linear map $A:\mathbb{R}^{d}\to\mathbb{R}^{d}$. For
$\lambda\in\mathbb{C}$, set

$E_{\lambda}:=\\{v\in\mathbb{C}^{d}:(A-\lambda I)^{j}v=0\text{\ for some\
}j\in\mathbb{N}-\\{0\\}\\}.$

The so-called stable, center and unstable subspaces $E^{u},E^{c},E^{s}$ are
the intersections with ${\mathbb{R}}^{d}$ of

$E^{u}_{\mathbb{C}}:=\bigoplus_{|\lambda|>1}E_{\lambda},\quad
E^{c}_{\mathbb{C}}:=\bigoplus_{|\lambda|=1}E_{\lambda},\quad
E^{s}_{\mathbb{C}}:=\bigoplus_{|\lambda|<1}E_{\lambda}$

respectively. Then $\mathbb{R}^{d}=E^{u}\oplus E^{c}\oplus E^{s}$. Denote
$E^{sc}=E^{s}\oplus E^{c}$. We can use the supremum norm $\|\cdot\|_{\infty}$
(denoted by $\|\cdot\|$ for short) on $\mathbb{R}^{d}$ defined by
$\|v\|:=\max\\{\|v^{u}\|,\|v^{c}\|,\|v^{s}\|\\}$ where
$v=v^{u}+v^{c}+v^{s},v^{*}\in E^{*},*\in\\{s,c,u\\}$. Denote
$B^{*}(\epsilon):=\\{v\in E^{*}:\|v\|<\epsilon\\},*\in\\{s,c,u\\}$.

Assume that $T_{A}:\mathbb{T}^{d}\to\mathbb{T}^{d}$ is ergodic. It is enough
to consider the nontrivial case $E^{u}\neq\emptyset$. The following lemma is
well known.

######  Lemma 5.10.

(Effective density, see Lemma 1 in [33]) Let $\tau>1$. Then for
$n\in{\mathbb{N}}$ large enough, $B^{u}(\tau^{n})$ is $(1/n^{d})$-dense in
$\mathbb{T}^{d}$.

It then follows form Lemma 5.10 that there exists $N_{1}\in{\mathbb{N}}$ such
that for any $n\geq N_{1}$, we have an integer function $M=M_{\delta}(n)$ such
that for any $x,x_{0}\in\mathbb{T}^{d}$,

$T_{A}^{M}(x+B^{u}(\delta/2))\cap\Big{(}x_{0}+B(\frac{\delta}{2n^{d-2}})\Big{)}\neq\emptyset$

and $\lim_{n\to\infty}\frac{M_{\delta}(n)}{n}=0$. Then
$T_{A}^{M}(x+B^{u}(\delta))$ contains a local unstable leaf
$W^{u}_{T_{A}}(y,2\delta)$ for some $y\in
W_{T_{A}}^{cs}(x_{0},\frac{\delta}{2n^{d-2}})$. We note that

  * •

$d(T_{A}^{n}x,T_{A}^{n}y)\leq\delta$ if $y\in
W_{T_{A}}^{cs}(x,\frac{\delta}{n^{d-2}})$,

  * •

$W^{cs}$ is a $C^{\infty}$ foliation,

  * •

$\lim_{n\to\infty}\frac{M_{\delta}(n)}{n}=0$.

The proof of Proposition 5.4 still works here with minor modifications and
thus Theorem E also follows from Theorem A.

We remark that Theorem E can be improved as follows.

######  Theorem E’.

Suppose that $T_{A}:\mathbb{T}^{d}\to\mathbb{T}^{d}$ is ergodic with
$E^{u}\neq\emptyset$. Then $h_{\mathop{\hbox{\footnotesize\rm
top}}}^{u}:C^{1}(\mathbb{T}^{d})\to{\mathbb{R}}^{+}\cup\\{0\\}$ is locally
constant at $T_{A}$, and $h_{\mathop{\hbox{\footnotesize\rm
top}}}:C^{1}(\mathbb{T}^{d})\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at
$T_{A}$.

###### Proof.

The first part of Theorem E’ can be derived from the proof of Theorem 1.1 in
[23]. The authors show that the unstable foliation of $T_{A}$ stably carries a
unique non-trivial homology and hence obtain the above result, taking into
account the fact $\chi_{u}(T_{A})=h_{\mathop{\hbox{\footnotesize\rm
top}}}^{u}(T_{A})$ by [20]. The second part of Theorem E’ can be proved
similarly as in the proof of Theorem D’ in Section 5.2.3. ∎

######  Remark 5.11.

The technique in [23] is inapplicable to homogeneous translations. Indeed, let
$g^{t},t\in{\mathbb{R}}$ be a one-parameter subgroup of $G$ with $g^{1}=g$,
then $T_{g}$ is isotopic to the identity map $T_{g^{0}}=T_{e}$ where $e$ is
the identity of the group $G$. So neither the stable foliation nor the
unstable foliation of $T_{g}$ carries a non-trivial homology.

###  5.4. Concluding remarks

In [46], the authors conjectured the topological entropy is continuous on the
space of $C^{1}$ PHDs with $1$-D center. They obtain the following list of
PHDs at which the topological entropy function is continuous in $C^{1}$
topology (except that item (7) is new).

  1. (1)

uniformly hyperbolic diffeomorphisms,

  2. (2)

skew products over uniformly hyperbolic diffeomorphisms, with the fiber being
a circle, and perturbations ([3, 37, 16, 17]),

  3. (3)

derived from Anosov diffeomorphisms ([32, 17, 38]),

  4. (4)

non-dynamically coherent examples of Hertz-Hertz-Ures ([44]),

  5. (5)

time-one maps of Anosov flows and perturbations ([22, 21, 46]),

  6. (6)

new examples by Bonatti-Parwani-Potrie ([2]) and Bonatti-Gogolev-Potrie ([1]),

  7. (7)

PHDs with uniformly compact center foliation of dimension one ([49]).

In this paper, we have investigated PHDs with higher dimensional center. Now
we can add to the above list the following PHDs at which the topological
entropy function $h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ and the unstable topological
entropy function $h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ are both lower semicontinuous:

  1. (8)

topologically transitive PHDs with bounded expansion in the center direction
(including time-one maps of frame flows; semisimple translations on homogenous
spaces; time-one maps of geodesic flows in symmetric spaces of noncompact
type, etc.),

  2. (9)

dynamically coherent PHDs which have subexponential growth in the center
direction and satisfy property (EM) (including translations on homogenous
spaces),

  3. (10)

ergodic toral automorphisms (see also [23]).

We also proved that the topological entropy function
$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{\infty}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at these
PHDs (if we assume PHDs in items (8) and (9) are $C^{\infty}$). The unstable
topological entropy function $h^{u}_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is always upper
semicontinuous. The topological entropy function
$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}:C^{1}(M)\to{\mathbb{R}}^{+}\cup\\{0\\}$ is continuous at item (10) and
translations on homogenous spaces in item (9).

Acknowledgement. This work is supported by NSFC Nos. 12071474 and 11701559.
The author would like to thank the referee for many helpful suggestions.

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