d_{g^{-1}\gamma}(f_{q(i)}^{e_{i}}\mu,f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu)$
$\displaystyle+$ $\displaystyle
d_{g^{-1}\gamma}(f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu,f_{q(i)}^{e_{i}}...f_{q(i+2)}^{e_{i+2}}\mu)+10$

We proceed to bound the terms of (7.1). Let the subsets
$\Psi_{a}^{\pm},\Delta_{a}^{\pm}$ and open subsets of $\mathcal{PML}(S)$
$U_{a}^{\pm}\supset\Delta_{a}^{\pm}$, $a=0,1,2,3$, be as in the proof of part
(i). The third term: $f_{q(i)}$ is supported on $X_{q(i)}$. If $g^{-1}\gamma$
is a boundary curve of $X_{q(i)}$ then by (7.3) this term is bounded above by
$2$. Otherwise, $A(g^{-1}\gamma)$ is not equal to $X_{q(i)}$, because $X_{a}$,
$a=0,1,2,3$, is a non-annular subsurface. Then applying Lemma 7.4 to the
curves in $\mu$ which overlap $A(g^{-1}\gamma)$, we get the upper bound
$m_{2}$ as was explained in the proof of part (i) where the subsurface was
non-annular. We may bound the second term by $m_{1}$ similarly. The term
fourth: If $g^{-1}\gamma$ is a boundary curve of $X_{q(i+1)}$ then by (7.3)
this term is bounded by $2$. Otherwise, $A(g^{-1}\gamma)$ is not equal to
$X_{q(i+1)}$. Then for $|e_{i}|>E_{2}$, $f_{q(i)}^{e_{i}}\mu\in
U_{q(i)}^{\pm}$, so we may get the bound $m_{2}$ as in part (i).

Thus we obtain the bound $6\max\\{2,m_{1},m_{2}\\}+2M+16$ for the sum (7.13).

In all of the other cases $d_{\gamma}(\mu_{I},\mu_{T})$ is bounded above by a
sum like (7.1) with at most $10$ terms. Then we may bound each term of the sum
as the above paragraph. Since the number of terms of the sum in each case is
at most $10$ (7.13) is bounded above by $10\max\\{2,m_{1},m_{2}\\}+2M+24$.

Case (2). Suppose that $i=3$. By Proposition 2.15,
$d_{\gamma}(\mu_{T},\partial{Z}_{3})\leq M$. Then by the triangle inequality

$d_{\gamma}(\mu_{I},\mu_{T})\leq
d_{\gamma}(\partial{Z}_{3},\mu_{T})+d_{\gamma}(\mu_{I},\partial{Z}_{3})\leq
d_{\gamma}(\mu_{I},\partial{Z}_{3})+M+1.$

Now $\mu\supset\\{\partial{X}_{a}\\}_{a=0,1,2,3}$ and
$\operatorname{diam}_{\gamma}(\mu)\leq 2$ so
$d_{\gamma}(\mu_{I},\partial{Z}_{3})\leq
d_{\gamma}(\mu,f_{q(1)}^{e_{1}}f_{q(2)}^{e_{2}}\mu)+2$. Then by the triangle
inequality we have

$d_{\gamma}(\mu_{I},\partial{Z}_{3})\leq
d_{\gamma}(\mu,f_{q(2)}^{e_{2}}\mu)+d_{\gamma}(f_{q(2)}^{e_{2}}\mu,f_{q(1)}^{e_{1}}f_{q(2)}^{e_{2}}\mu)+2+2$

The first term above can be bounded as the third term of (7.13) by $2$ or
$m_{1}$. The second term can be bounded as the fourth term of (7.13) by $2$ or
$m_{2}$. So $d_{\gamma}(\mu_{I},\mu_{T})\leq 2\max\\{2,m_{1},m_{2}\\}+M+5$.
When $i=5$, with a similar argument we can obtain the bound
$d_{\gamma}(\mu_{I},\mu_{T})\leq 4\max\\{2,m_{1},m_{2}\\}+M+9$.

Case (3). Considering $q^{\prime}(i)=q(k-i+1)$ for $i=1,...,k$ and
$e^{\prime}_{i}=e_{k-i+1}$ for $i=1,...,k$ the argument of Case (2) for
$q^{\prime}$ and $e^{\prime}_{i}$ gives us the bound
$d_{\gamma}(\mu_{I},\mu_{T})\leq 5\max\\{2,m_{1},m_{2}\\}+M+9$.

Establishing the bounds in cases (1), (2) and (3) we conclude that
$\textbf{m}^{\prime}=10\max\\{2,m_{1},m_{2}\\}+4M+24$ is the desired bound in
(ii). ∎

######  Proposition 7.11.

There are $K_{1}\geq 1$ and $C_{1}\geq 0$ and $E>E_{2}$, depending only on the
partial pseudo-Anosov maps $f_{0},f_{1},f_{2}$ and $f_{3}$ and $\mu_{I}$ with
the following properties. Given $q$ and $\\{e_{i}\\}_{i}$ such that
$|e_{i}|>E$ for any $i\in\\{1,...,k\\}$, we have

(7.15) $d_{Z_{i}(q,e)}(\mu_{I}(q,e),\mu_{T}(q,e))\asymp_{K_{1},C_{1}}|e_{i}|$

###### Proof.

By Lemma 7.6, there are $K^{\prime}_{1},C^{\prime}_{1}$ and $E_{1}$, such that
when $|e_{i}|>E_{1}$ for all $i\in\\{1,...,k\\}$, we have
$d_{Z_{i}}(\mu_{I},\mu_{T})\geq K^{\prime}_{1}|e_{i}|-C^{\prime}_{1}$.

Let $\mu=\mu_{I}(q,e)$. Let $i\in\\{1,...,k\\}$ and
$g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$. Applying $g^{-1}$,
$d_{Z_{i}}(\mu,\mu_{T})=d_{X_{q(i)}}(g^{-1}\mu,f_{q(i)}^{e_{i}}h\mu)$, where
$h=f_{q(i+1)}^{e_{i+1}}...f_{q(k)}^{e_{k}}$. Now by the triangle inequality

$d_{X_{q(i)}}(g^{-1}\mu,f_{q(i)}^{e_{i}}h\mu)\leq
d_{X_{q(i)}}(f_{q(i)}^{e_{i}}h\mu,h\mu)+d_{X_{q(i)}}(h\mu,g^{-1}\mu).$

In Lemma 7.6, we proved that $d_{X_{q(i)}}(h\mu,g^{-1}\mu)\leq
C^{\prime}_{1}$. So here we only need to show that for some $K_{1}\geq 1$,
$d_{X_{q(i)}}(f_{q(i)}^{e_{i}}h\mu,h\mu)\geq K_{1}|e_{i}|$.

Define $q^{\prime}(j)=q(j+i-1)$ for $j=1,...,k-i+1$ and
$e^{\prime}_{j}=e_{j+i-1}$ for $j=1,...,k-i+1$. Then the subsurface $X_{q(i)}$
is not in the list of subsurfaces $Z_{j}(q^{\prime},e^{\prime})$. So by
Proposition 7.9 (i) there are $E_{2}$ and m such that: if $|e_{i}|>E_{2}$ then
$d_{X_{a}}(\mu,h\mu)\leq\textbf{m}$. Note that this bound does not depend on
$e_{i}$.

Let
$\bar{\tau}_{a}:=\bar{\tau}_{f_{a}}=\limsup_{n\to\infty}\frac{d_{X_{a}}(\delta,f^{n}\delta)}{n}$
for every $\delta\in\mathcal{C}(X_{a})$, $a=0,1,2,3$, as in Lemma 7.2. Let
$a\in\\{0,1,2,3\\}$. Given $\delta\in\mathcal{C}(X_{a})$, there is
$N=N(\delta)$ such that if $n\geq N$, then
$d_{X_{a}}(\delta,f^{n}\delta)\leq(\bar{\tau}_{a}+1)n$. Moreover, note that
for $\delta^{\prime}\neq\delta$, $|N(\delta)-N(\delta^{\prime})|$ is bounded
by a constant depending only on the distance of $\delta$ and $\delta^{\prime}$
in $\mathcal{C}(X_{a})$ (see the proof of Lemma 7.2). Thus $N$ is the same for
all $\delta$ in the $\textbf{m}-$neighborhood of
$\pi_{X_{a}}(\mu)\in\mathcal{C}(X_{a})$. Then there exists $E>E_{2}$ (recall
that $E_{2}>E_{1}$) such that for any $n\geq E$,
$d_{X_{a}}(\delta,f_{a}^{n}(\delta))\leq(\bar{\tau}_{a}+1)n$ for every curve
$\delta$ in the $\textbf{m}-$neighborhood of $\pi_{X_{a}}(\mu)$ in
$\mathcal{C}(X_{a})$ and $a=0,1,2,3$. Now since $\pi_{X_{q(i)}}(h\mu)$ is in
the $\textbf{m}-$neighborhood of $\pi_{X_{q(i)}}(\mu)$, if $|e_{i}|>E$, we
have $d_{X_{q(i)}}(h\mu,f_{q(i)}^{e_{i}}h\mu)\leq(\bar{\tau}_{a}+1)|e_{i}|$.

Let $K_{1}=\max\\{\bar{\tau}_{a}+1,\frac{1}{\tau_{a}}:a=0,1,2,3\\}$ then as we
saw above $d_{X_{q(i)}}(\mu,h\mu)\leq K_{1}|e_{i}|$. Further, by Lemma 7.6 (i)
$d_{X_{q(i)}}(\mu,h\mu)\geq
K_{1}^{\prime}|e_{i}|-C_{1}\geq\frac{1}{K_{1}}|e_{i}|-C^{\prime}_{1}$. Thus
for constants $K_{1},C_{1}=C^{\prime}_{1}$ and $E$ the proposition holds.

∎

Lemma 7.6 and propositions 7.9 and 7.11 together prescribe the list of all
subsurface coefficients of a hierarchy path between $\mu_{I}(q,e)$ and
$\mu_{T}(q,e)$ (corresponding to $q$ and $\\{e_{i}\\}_{i}$ ). To generalize
this construction to infinite hierarchy paths, let
$q:\mathbb{N}\to\\{0,1,2,3\\}$ be as the beginning of this subsection and let
$e_{i}$ be a sequence of integers. For each $i\geq 1$ set the subsurface
$Z_{i}(q,e)=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}\partial{X}_{q(i)}$. Define
$q^{k}:\\{1,...,k\\}\to\\{0,1,2,3\\}$ by $q^{k}(i)=q(i)$ for $i=1,...,k$, and
define the sequence $e^{k}_{i}=e_{i}$ for $i=1,...,k$. Let
$\mu\equiv\mu_{I}(q^{k},e^{k})$ be a marking containing
$\\{\partial{X}_{a}\\}_{a=0,1,2,3}$ and let $\mu_{k}=\mu_{T}(q^{k},e^{k})$ be
defined as before. For each $k\geq 1$, let $\delta_{k}$ be a curve in the base
of the marking $\mu_{k}$, after possibly passing to a subsequence
$\delta_{k}$’s converge to a lamination $\lambda$ in the Hausdorff topology of
closed subsets of the surface $S$. Note that by Proposition 2.4 $\lambda$
contains the support of any accumulation point of the projective classes
$[\delta_{k}]$ in the $\mathcal{PLM}(S)$ topology. Here each $\delta_{k}$ is
equipped with the transversal measure $i(\delta_{k},.)$.

######  Proposition 7.12.

There are constants $\textbf{m},\textbf{m}^{\prime}>4M$, $C_{1}\geq 0$,
$K_{1}\geq 1$, depending only on $f_{0},f_{1},f_{2}$ and $f_{3}$, and
$\mu_{I}$ with the following properties. Given $q$ and $\\{e_{i}\\}_{i}$ such
that $|e_{i}|\geq E$ for any $i$ in the domain of $q$, we have

  1. (i)

For any integer $i$ in the domain of $q$,
$d_{Z_{i}(q,e)}(\mu_{I}(q,e),\mu_{T}(q,e))\asymp_{K_{1},C_{1}}|e_{i}|$.

  2. (ii)

For any non-annular subsurface $W$ which is neither $Z_{i}$ for some $i$ nor
$S$ we have $d_{W}(\mu_{I},\mu_{T})\leq\textbf{m}$.

  3. (iii)

For any $\gamma\in\mathcal{C}_{0}(S)$ we have
$d_{\gamma}(\mu_{I},\mu_{T})\leq\textbf{m}^{\prime}$.

  4. (iv)

Given $i,j\geq 1$, if $j\geq i+2$, then $Z_{i}<Z_{j}$.

  5. (v)

Let $i$ be such that $q(i)=1$ or $3$. If $j\in J_{Z_{i}}$ then $j\geq\min
J_{Z_{i-1}}$ and $j\leq\max J_{Z_{i+1}}$.

When the domain of $q$ is $\mathbb{N}$,

  6. (vi)

$\mu_{T}(q,e)$ is a minimal filling lamination.

###### Proof.

When the domain of $q$ is $\\{1,...,k\\}$ for some $k\geq 1$ the subsurface
coefficient bounds (ii) and (iii) are already established Proposition 7.9 and
(i) in Propositions 7.11. Further, the order of domains (iv) is established in
Lemma 7.6 (ii) and (v) is proved in Lemma 7.8.

We proceed to establish the bounds when the domain of $q$ is $\mathbb{N}$.
Given $i\geq 1$, let $Z_{i}=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}$.
By Proposition 7.11 we have that for every $k\geq i$,
$d_{Z_{i}}(\mu,\mu_{k})\asymp_{K_{1},C_{1}}|e_{i}|$. Then the Hausdorff
convergence of $\delta_{k}$’s to $\mu_{T}(q,e)$ ($\delta_{k}$ is a curve in
the base of $\mu_{k}$) implies that for all $k$ sufficiently large
$d_{Z_{i}}(\mu,\mu_{T})\asymp_{1,1}d_{Z_{i}}(\mu,\mu_{k})$. (i) is
established.

Let $W$ be a subsurface which is neither $Z_{i}$ for some $i\geq 1$ nor $S$.
By Proposition 7.9 (i) for any $k\geq 1$ we have that
$d_{W}(\mu,\mu_{k})\leq\textbf{m}$. Then the Hausdorff convergence of
$\delta_{k}$’s to $\mu_{T}(q,e)$ implies that for all $k$ sufficiently large
$d_{W}(\mu,\mu_{T})\asymp_{1,1}d_{W}(\mu,\mu_{k})$. (ii) is established.
Similarly Proposition 7.9 (ii) and the Huasdorff convergence imply that
$d_{\gamma}(\mu,\mu_{k})\leq\textbf{m}^{\prime}$, (iii) is established.

Given $k\geq 1$ and $i,j\leq k$ by Lemma 7.6 (ii) the order of the subsurfaces
$Z_{i}$ and $Z_{j}$ with respect to $\mu$ and $\mu_{k}$ is: $Z_{i}<Z_{j}$ if
$j\geq i+2$. By Proposition 2.15 this means that $Z_{i}\pitchfork Z_{j}$ and
the subsurface coefficient bounds $d_{Z_{i}}(\mu,\mu_{k})>4M$,
$d_{Z_{j}}(\mu,\mu_{k})>4M$ and $d_{Z_{i}}(\mu,\partial{Z}_{j})>2M$ hold. Then
the Hausdorff convergence of $\mu_{k}$ to $\mu_{T}$ implies that the first two
subsurface coefficient lower bounds hold for $\mu$ and $\mu_{T}(q,e)$ and (iv)
follows for $\mu$ and $\mu_{T}(q,e)$.

The proof of (v) in Lemma 7.8 uses the fact that $Z_{i}$ subsurface
coefficients are greater than or equal to $2M$, and the way that the
subsurfaces $Z_{i-1}$ and $Z_{i}$, and $Z_{i}$ and $Z_{i+1}$ intersect. By
part (i) the $Z_{i}$ subsurface coefficients of $\mu$ and $\mu_{T}$ are
greater than $2M$. Moreover, their pattern of intersection is the same as the
lemma. Then the proof of Lemma 7.8 goes through and gives us (v) when the
domain of $q$ is $\mathbb{N}$ holds as well.

For each $k\geq 1$ let $\rho_{k}$ be a hierarchy path between $\mu$ and
$\mu_{k}$. Let $\rho$ be a hierarchy path between $\mu$ and $\mu_{T}(q,e)$.
For any $i$ with $q(i)=1$ or $3$, by part (i),
$d_{Z_{i}}(\mu,\mu_{T}(q,e))>M$, so $Z_{i}$ is a component domain of $\rho$
with one boundary curve. Thus $\partial{Z}_{i}$ is a curve on the main
geodesic of $\rho$. Furthermore, $Z_{i}<Z_{j}$ implies that $\partial{Z}_{i}$
is before $\partial{Z}_{j}$ along the main geodesic of $\rho$. To see this,
note that since $\partial{Z}_{i}$ and $\partial{Z}_{j}$ are on the main
geodesic and each consist of one curve, the tight geodesics $g_{Z_{i}}$ and
$g_{Z_{j}}$ are time ordered as is defined in $\S 4$ of [MM00]. Then if
$\partial{Z}_{i}$ is after $\partial{Z}_{j}$ there would be an $m\in
J_{Z_{i}}$ so that $m>\max J_{Z_{j}}$ (see $\S 5$ of [MM00]), but this
contradicts the second bullet of Proposition 2.15. Therefore, the curves
$\partial{Z}_{i}$ converge to a point $\xi$ in the Gromov boundary of
$\mathcal{C}(S)$ as $i\to\infty$. This point by Theorem 2.2 determines a
projective measured lamination $[\mathcal{E}]$ with minimal filling support.
Furthermore, given $k\geq 1$, for each $1\leq i\leq k$,
$d_{Z_{i}}(\mu,\mu^{k})>M$, so $\partial{Z}_{i}$ is on the main geodesic of
$\rho_{k}$ as well. This implies that as $k\to\infty$,
$\delta_{k}\in\operatorname{base}(\mu_{k})$ converge to $\xi$ in the Gromov
boundary of $\mathcal{C}(S)$. By Theorem 2.3 the projective classes
$[\delta_{k}]$ after possibly passing to a subsequence converge to a
lamination with support equivalent to the support of $\mathcal{E}$. Then by
Proposition 2.4 . is contained in $\mu_{T}(q,e)$ which implies that
$\mu_{T}(q,e)$ is minimal filling.

∎

###  7.2. Scheme II

The construction of this subsection will be used in $\S$8.4 to provide
examples of recurrent WP geodesic rays.

Let $\alpha$ be a curve such that $S\backslash\alpha$ is a large subsurface.
Consider the indexed subsurfaces $X_{0}=S$ and $X_{1}=S\backslash\alpha$. Let
$f_{0},f_{1}$ be partial pseudo-Anosov maps supported on $X_{0}$ and $X_{1}$,
respectively. Define functions $q_{0}(i)\equiv i$ (mod 2) and
$q_{1}(i)=q_{0}(i+1)$. Let $q$ be any of $q_{0}$ and $q_{1}$ or their
restriction to $\\{1,...,k\\}$, where $k$ is a positive integer.

Let $q$ be as above. When the domain of $q$ is $\mathbb{N}$ let
$\\{e_{i}\\}_{i}$ be an infinite sequence of integers and when the domain of
$q$ is $\\{1,...,k\\}$ let $\\{e_{i}\\}_{i}$ be a sequence of integers with
$k$ elements.

For any $i$ in the domain of $q$ set the subsurface

$Z_{i}(q,e)=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}$

Let $\mu_{I}(q,e)$ be a marking whose base contains
$\\{\partial{X}_{a}\\}_{a=0,1}=\\{\alpha\\}$. When the domain of $q$ is
$\\{1,...,k\\}$ for some $k\geq 1$ let
$\mu_{T}(q,e)=f_{q(1)}^{e_{1}}...f_{q(k)}^{e_{k}}\mu_{I}(q,e)$. When the
domain of $q$ is $\mathbb{N}$ define $q^{k}(i)=q(i)$ for $i=1,..,k$ and the
sequence $e^{k}_{i}=e_{i}$ for $i=1,...,k$. Let $\delta_{k}$ be a curve in the
base of $\mu_{T}(q^{k},e^{k})$, after possibly passing to a subsequence
$\delta_{k}$ converge to a lamination $\mu_{T}(q,e)$ in the a Hausdorff
topology of closed subsets of the surface.

In the following proposition we establish several bounds on the subsurface
coefficients of $\mu_{I}(q,e)$ and $\mu_{T}(q,e)$.

######  Remark 7.13.

The construction of this subsection and the estimates on subsurface
coefficients can be carried out in a more general setting. Here we restrict
ourself to be able to provide detailed step by step estimates and complete
arguments.

######  Proposition 7.14.

There are constants $\textbf{m},\textbf{m}^{\prime}>4M$, $C_{1}\geq 0$ and
$K_{1}\geq 1$ depending only on $f_{0},f_{1}$ and $\mu_{I}$ with the following
properties. Given $q$ and $\\{e_{i}\\}_{i}$ such that $|e_{i}|>E$ for any $i$
in the domain of $q$, we have

  1. (i)

For any integer $i$ in the domain of $q$ with $q(i)=1$,

$d_{Z_{i}(q,e)}(\mu_{I}(q,e),\mu_{T}(q,e))\asymp_{K_{1},C_{1}}|e_{i}|.$

  2. (ii)

Given $i,j\geq 1$, if $i<j$ and $q(i)=q(j)=0$ then $Z_{i}<Z_{j}$. If $j<i$ and
$q(i)=q(j)=0$ then $Z_{j}<Z_{i}$.

  3. (iii)

For any non-annular subsurface $W$ which is neither $Z_{i}$ for some $i$ nor
$S$ we have $d_{W}(\mu_{I},\mu_{T})\leq\textbf{m}$.

  4. (iv)

For any $\gamma\in\mathcal{C}_{0}(S)$ we have
$d_{\gamma}(\mu_{I},\mu_{T})\leq\textbf{m}^{\prime}$.

When the domain of $q$ is $\mathbb{N}$ we have

  5. (v)

$\mu_{T}(q,e)$ is a minimal filling lamination.

###### Proof.

First we prove the statements (i) to (iv) when for some $k\geq 1$,
$q:\\{1,...,k\\}\to\\{0,1\\}$ and $\\{e_{i}\\}_{i=1}^{k}$ is a sequence of
integers with $k$ elements. Most of the details are similar to the ones given
in $\S$ 7.1 (Scheme I) so here we mainly sketch them and explain the necessary
modifications.

We set the constants: $K^{\prime}_{1}=\min_{a=0,1}\\{\tau_{a}\\}$ where the
constant $\tau_{a}:=\tau_{f_{a}}$ is from the Proposition 7.1,
$C^{\prime}_{1}=2(B_{0}+\eta)$, where
$\eta=\max\\{d_{X_{a}}(f_{b}^{e}\mu,\mu):a,b\in\\{0,1\\}\;\text{and }\;a\neq
b\\}$, and $E_{1}=\frac{B_{0}+4+4M+\omega+C_{1}^{\prime}}{K^{\prime}_{1}}$,
where $\omega=\max\\{d_{W}(\mu,\partial{X_{a}}):W\subseteq
S\;\text{and}\;a=0,1\\}$.

Let $l\in\\{1,...,k\\}$ be such that $q(l)=1$. Let
$g=f_{q(1)}^{e_{1}}...f_{q(l-1)}^{e_{l-1}}f_{q(l)}^{e_{l}}$. $f_{q(l)}$ is
supported on $S\backslash\alpha$ and preserves $\alpha$ and by the definition
of $q$, $X_{q(l+2)}=S\backslash\alpha$. Then applying $g^{-1}$ to
$d_{S}(\partial{Z}_{l},\partial{Z}_{l+2})$ we get
$d_{S}(\alpha,f_{q(l+1)}^{e_{l+1}}\alpha)$. $f_{q(l+1)}=f_{0}$ is supported on
$S$, so by Proposition 7.1,
$d_{S}(\alpha,f_{q(l+1)}^{e_{l+1}}\alpha)\geq\tau_{0}|e_{l+1}|>\tau_{0}E_{1}\geq
3$. Therefore,

(7.16) $d_{S}(\partial{Z}_{l},\partial{Z}_{l+2})\geq\tau_{0}|e_{l+1}|.$

Let $i\in\\{1,...,k\\}$ be such that $q(i)=1$. Then the proof of Lemma 7.6 (i)
goes through line by line and gives us

(7.17) $d_{Z_{i}(q,e)}(\mu_{I}(q,e),\mu_{T}(q,e))\geq
K^{\prime}_{1}|e_{i}|-C^{\prime}_{1}$

The only difference is that Claim 7.7 which asserts that
$\partial{X}_{q(i)}\pitchfork f_{q(i+1)}^{e_{i+1}}X_{q(i+2)}$ here is proved
as follows: $\partial{X}_{q(i)}=\alpha$ and by the definition of $q$,
$\partial{X}_{q(i+2)}=\alpha$. Then as we saw above
$d_{S}(\alpha,f_{q(i+1)}^{e_{i+1}}(\alpha))\geq 3$ which implies that
$\alpha\pitchfork f_{q(i+1)}^{e_{i+1}}\alpha$.

Then as part (ii) of Lemma 7.6 we may show that the subsurfaces with $q(i)=1$
are ordered. Here we establish the base of induction as follows: By (7.16)
sicne $|e_{2}|>E_{1}$ we have that $d_{S}(\partial{Z}_{1},\partial{Z}_{3})\geq
3$ and thus $\partial{Z}_{1}\pitchfork\partial{Z}_{3}$. Further by (7.17)
$d_{Z_{1}}(\mu_{I},\partial{Z}_{3})>2M$. Then by Proposition 2.15
$Z_{1}<Z_{3}$.

Proof of part (iii). Suppose that $d_{W}(\mu_{I},\mu_{T})>4M$ (otherwise we
already have the bound). Let $l$ be such that $q(l)=1$. Since
$|e_{l+1}|>E_{1}$, it follows from (7.16) that $\partial{Z_{l}}$ and
$\partial{Z_{l+2}}$ fill $S$. Thus $W$ is ordered with respect to either
$Z_{l}$ or $Z_{l+2}$. If $Z_{l}<W<Z_{l+2}$ then we are in case (1) below.
Otherwise, either $Z_{l}<Z_{l+2}<W$ or $W<Z_{l}<Z_{l+2}$. In the former
situation compare $W$ with $Z_{l+2}$ and $Z_{l+4}$ and in the later situation
with $Z_{l}$ and $Z_{l-2}$. Repeating this comparison we may conclude that $W$
is ordered in the list of subsurfaces $Z_{i}$ as one of the following cases.

  1. (1)

There is an index $i$ such that either $Z_{i-1}<W<Z_{i+1}$ or
$Z_{i-2}<W<Z_{i+2}$,

  2. (2)

$W<Z_{i}$, where $i=1$ or $2$,

  3. (3)

$Z_{i}<W$, where $i=k-1$ or $k-2$.

First suppose that $Z_{i-2}<W<Z_{i+2}$. Then

$d_{W}(\mu_{I},\mu_{T})\leq d_{W}(\partial{Z}_{i-2},\partial{Z}_{i+2})+2M.$

Let $g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$. Applying $g^{-1}$ to the
subsurface on the left hand side we obtain
$d_{g^{-1}W}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\partial{X}_{q(i-2)},f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\partial{X}_{q(i+2)})$.
Recall that $\mu_{I}\supset\\{\partial{X}_{a}\\}_{a=0,1,2,3}$. Denote
$\mu_{I}$ by $\mu$. Since $\operatorname{diam}_{g^{-1}W}(\mu)\leq 2$,

$d_{g^{-1}W}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\mu,f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu)+4$

bounds the last subsurface coefficient. By the triangle inequality this is
bounded above by

(7.18) $\displaystyle
d_{g^{-1}W}(f_{q(i-1)}^{-e_{i-1}}f_{q(i-2)}^{-e_{i-2}}\mu,f_{q(i-2)}^{-e_{i-2}}\mu)$
$\displaystyle+$ $\displaystyle
d_{g^{-1}W}(f_{q(i-2)}^{-e_{i-2}}\mu,\mu)+d_{g^{-1}W}(\mu,f_{q(i)}^{e_{i}}\mu)$
$\displaystyle+$ $\displaystyle
d_{g^{-1}W}(f_{q(i)}^{e_{i}}\mu,f_{q(i)}^{e_{i}}f_{q(i+1)}^{e_{i+1}}\mu)+6$

######  Claim 7.15.

For any $a\in\\{0,1\\}$, $X_{a}\neq g^{-1}W$, where
$g=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}$.

First suppose that $X_{q(i)}=S\backslash\alpha$.

  * •

$g^{-1}W=S\backslash\alpha$. Then $W=gX_{q(i)}=Z_{i}$, but this contradicts
the fact that $W$ is not in the list of $Z_{i}$’s.

  * •

$g^{-1}W=S$. Then $W=S$, but this contradict the fact that $W$ is a proper
subsurface.

Now suppose that $X_{q(i)}=S$.

  * •

$g^{-1}W=S\backslash\alpha$. $f_{q(i+1)}$ is supported on $S\backslash\alpha$
and preserves $\alpha$. Then
$W=gS\backslash\alpha=gf_{q(i-1)}^{-e_{i-1}}S\backslash\alpha=Z_{i-1}$, but
this contradicts the fact that $W$ is not in the list of $Z_{i}$’s.

  * •

$g^{-1}W=S$ contradicts the fact that $W$ is a proper subsurface.

These contradictions show that $X_{a}\neq g^{-1}W$. The claim is proved.

Let $\Psi_{a}^{\pm}=\Psi_{f_{a}}^{\pm}$ and
$\Delta_{a}^{\pm}=\Delta_{f_{a}}^{\pm}$, $a=0,1$, be the subsets of
$\mathcal{PML}(S)$ as in the beginning of $\S$7. Fix neighborhoods
$U_{a}^{\pm}\supset\Delta_{a}^{\pm}$ such that $\overline{U_{a}^{-}}$ and
$\overline{U_{a}^{+}}$ be disjoint from $\Psi^{-}_{b}\cup\Psi_{b}^{+}$ for any
$a,b\in\\{0,1\\}$ with $a\neq b$. Applying Lemma 7.4 to the pseudo-Anosov map
$f_{a}$ and compact sets $\pi_{X_{b}}(\mu)$, $\overline{U_{b}^{-}}$ and
$\overline{U_{b}^{+}}$, for any $a,b\in\\{0,1\\}$ with $a\neq b$, there exists
$E_{2}>E_{1}$ such that

  * •

$f_{a}^{n}(\pi_{X_{b}}(\mu))\subset U_{b}^{+}$ for all $n\geq E_{2}$ and
$f_{a}^{-n}(\pi_{X_{b}}(\mu))\subset U_{b}^{-}$ for all $n\geq E_{2}$, and

  * •

$f_{a}^{n}\overline{U_{a}^{\pm}}\subset U_{b}^{+}$ for all $n\geq E_{2}$ and
$f_{a}^{-n}\overline{U_{a}^{\pm}}\subset U_{a}^{-}$ for all $n\geq E_{2}$.

Now using Claim 7.15 we may proceed as in the proof of part (i) of Proposition
7.9 and bound all of the terms in (7.18) by either
$m_{1}=\max_{a=0,1}m(f_{a},\mu)$ or
$m_{2}=\max_{a=0,1}m(f_{a},\overline{U_{a}^{\pm}})$. Then
$4\max\\{m_{1},m_{2}\\}+2M+16$ is an upper bound for $d_{W}(\mu_{I},\mu_{T})$.
Moreover we may get the bound $m_{1}+M+5$ for $d_{W}(\mu_{I},\mu_{T})$ in
cases (2) and (3) following the argument in Proposition 7.9. Thus
$\textbf{m}=4\max\\{m_{1},m_{2}\\}+4M+16$.

Proof of part (iv). By (7.16) $d_{S}(\partial{Z}_{l-2},\partial{Z}_{l})\geq 3$
so $\partial{Z}_{l}$ and $\partial{Z}_{l-2}$ fill $S$. Similarly
$d_{S}(\partial{Z}_{l+2},\partial{Z}_{l})\geq 3$ and $\partial{Z}_{l}$ and
$\partial{Z}_{l+2}$ fill $S$. Thus

  * •

$\gamma\pitchfork\partial{Z}_{l}$ or $\gamma\pitchfork\partial{Z}_{l-2}$, and

  * •

$\gamma\pitchfork\partial{Z}_{l}$ or $\gamma\pitchfork\partial{Z}_{l+2}$

Then similar to the proof of part (ii) of Proposition 7.9 we can show that
$\gamma$ is ordered in the list of $Z_{i}$’s as one of the following cases:

  1. (1)

There is an index $i$ such that either $Z_{i-1}<W<Z_{i+1}$ or
$Z_{i-2}<W<Z_{i+2}$,

  2. (2)

$\mu_{I}<\gamma<Z_{i}$, where $i=1$ or $2$,

  3. (3)

$Z_{i}<\gamma<\mu_{T}$, where $i=k-1$ or $k-2$.

Then similar to the proof of part (ii) of Proposition 7.9 we may establish the
upper bound $\textbf{m}^{\prime}=4\max\\{2,m_{1},m_{2}\\}+4M+16$ for
$d_{\gamma}(\mu_{I},\mu_{T})$.

The lower bound (7.17) and part (iii) together as in the proof of Proposition
7.11 give us (i). Here we set $E>E_{2}$ such that for any integer $e$ with
$|e|>E$, $d_{X_{a}}(\delta,f_{a}^{e}(\delta))\leq(\bar{\tau}_{a}+1)|e|$ for
every $\delta$ in the $\textbf{m}-$neighborhood of $\pi_{X_{a}}(\mu)$ in
$\mathcal{C}(X_{a})$ and $a=0,1$. Then for
$K_{1}=\max\\{\frac{1}{\tau_{a}},\bar{\tau}_{a}+1\\}$, $C_{1}=C^{\prime}_{1}$
and $E$ (i) holds.

Establishing all of the bounds when the domain of $q$ is $\\{1,...,k\\}$ for
some $k\geq 1$, the bounds when the domain of $q$ is $\mathbb{N}$ and the fact
that $\mu_{T}$ is minimal filling, part (v), follow from the limiting argument
we gave in the proof of Proposition 7.12. The order of subsurface, part (ii),
also follows from the one when the domain of $q$ is finite we established
above and the limiting argument we gave in the proof of Proposition 7.12.

∎

##  8\. Weil-Petersson Geodesics

In this section we use the control on length-functions along WP geodesic
segments from $\S$6 and the pair of laminations/markings with prescribed list
of subsurface coefficients from $\S$7 to provide examples of Weil-Petersson
geodesics with certain behavior in the moduli space. For example divergent
rays in the moduli space and closed geodesics in the thin part of moduli
space. We also provide a recurrence condition for WP geodesics in terms of
ending laminations. Our results in this section can be considered as a kind of
symbolic coding of WP geodesics.

In [PWW10], the authors construct WP geodesics which are dense in the moduli
space when $\xi(S)=1$. Jeff Brock produces examples of divergent WP geodesic
rays with minimal filling ending lamination in any complexity. Both
constructions start with a piecewise geodesic in the Weil-Petersson completion
of Teichmüller space. Then applying high Dehn twists about curves in the
multi-curves which determine the strata that the piecewise geodesic intersects
manage to replace the piecewise geodesic with a piecewise geodesic in the
completion of Teichmüller space with arbitrary small exterior angles. Then
using a kind of shadowing lemma perturb it to a single WP geodesic in the
Teichmüller space. In these examples the relation between the itinerary of the
ray the end invariants and their associated subsurface coefficients is not
explicit.

These constructions are analogue of the ones for Teichmüller geodesics. Cheung
and Masur in [CM06] give examples of divergent Teichmüller geodesic rays with
uniquely ergodic vertical lamination. Rafi using the control on length
functions in terms of subsurface coefficients he developed along Teichmüller
geodesic constructs closed Teichmüller geodesics staying in the thin part and
divergent geodesic rays.

###  8.1. Weil-Petersson geodesic rays with prescribed itinerary

In this subsection by extracting limits of WP geodesic segments with end
invariants on a single infinite hierarchy path with narrow end points we
construct WP geodesics whose behavior mimic the combinatorial properties of
hierarchy paths.

Let $\nu\in\mathcal{GL}(S)$. Let $Z_{a}$, $a=1,...,m$, be the connected
components of $S\backslash\\{\text{closed leaves of}\;\nu\\}$ with
$\xi(Z_{a})\geq 1$ such that $\nu_{a}$ the restriction of
$\nu\backslash\\{\text{closed leaves of}\;\nu\\}$ to $Z_{a}$ is minimal
filling on $Z_{a}$. For $a=1,...,m$ fix measures $\mathcal{L}_{a}$ supported
on $\nu_{a}$. By Theorem 2.2 the projective class of the measure
$\mathcal{L}_{a}$ supported on $\nu_{a}$ is a point in the Gromov boundary of
$\mathcal{C}(Z_{a})$. Let $\gamma_{n,a}\in\mathcal{C}(Z_{a})$ be a sequence of
curves which converges to $\mathcal{L}_{a}$ as $n\to\infty$ in the weak∗
topology of $\mathcal{ML}(Z_{a})$. For $n\geq 1$ let $Q_{n}$ be a pants
decomposition containing $\bigcup_{a=1}^{m}\partial{Z}_{a}\cup\gamma_{n,a}$.
Let $[x,c_{n}]$ be the WP geodesic segment connecting a base point $x$ in the
interior of Teichmüller space to $c_{n}$, a maximally nodal hyperbolic surface
at $Q_{n}$. Denote the parametrization of $[x,c_{n}]$ by arc-length by
$r_{n}$. The proof of the following lemma essentially follows the proof of
surjectivity of weighted ending laminations of WP geodesic rays in the
Teichmüller space of surface with complexity $5$ given in $\S 4$ of [BM08].

######  Lemma 8.1.

(Infinite ray) After possibly passing to a subsequence the geodesic segments
$r_{n}$ converge to an infinite ray $r$ in the Weil-Petersson visual sphere at
$x$. Furthermore the forward ending lamination of $r$ contains $\nu_{a}$, for
all $a=1,...,m$, and the length of every curve $\alpha\in\partial{Z}_{a}$,
$a=1,...,m$, is decreasing along $r$.

###### Proof.

By Theorem 3.2 (Non-refraction Theorem) the interior of each one of the WP
geodesic segments $r_{n}$ is inside the Teichmüler space. Moreover, by the
local compactness of the WP metric at $x$ after possibly passing to a
subsequence the initial parts of $r_{n}$’s converge to a geodesic segment
$r_{\infty}$ starting at $x$. Let $r$ be the maximal geodesic ray in
Teichmüller space with initial part $r_{\infty}$. We prove that $r$ is an
infinite ray.

Let $s_{n}=\frac{1}{\ell_{Q_{n}}(x)}$, where $\ell_{Q_{n}}(x)=\max_{\gamma\in
Q_{n}}\ell_{\gamma}(x)$. Then for each integer $n$, and $t$ in the domain of
$r_{n}$ we have that $\ell_{s_{n}Q_{n}}(r_{n}(t))\leq 1$. This follows from
the convexity of length-functions along WP geodesics and the observation that
$\ell_{s_{n}Q_{n}}(x)=1$ and $\ell_{s_{n}Q_{n}}(c_{n})=0$. Now assuming that
$r(t)$ has finite length $T$ we get a contradiction.

After possibly passing to a subsequence $s_{n}Q_{n}$ converge to some
$\mathcal{L}\in\mathcal{ML}(S)$ in the weak∗ topology of $\mathcal{ML}(S)$.
Let $\mathcal{L}_{a}$ be the restriction of $\mathcal{L}$ to $Z_{a}$,
$a=1,...,m$. By the way we chose $Q_{n}$’s, the projective class of
$\pi_{Z_{a}}(Q_{n})$ converge to the projective class of a measure supported
on $\nu_{a}$. Then by Theorem 2.3 the support of $\mathcal{L}_{a}$ is a
measured lamination equivalent to $\nu_{a}$ and in particular fills $Z_{a}$.
Now let $\gamma$ be a curve with $i(\gamma,\mathcal{L})=0$. Then it is
disjoint from $\mathcal{L}_{a}$’s and since for each $a$, $\mathcal{L}_{a}$
fills $Z_{a}$, $\gamma$ is disjoint from all $Z_{a}$’s.

The length-function
$\ell_{\mathcal{L}}(x):\operatorname{Teich}(S)\times\mathcal{ML}(S)\to\mathbb{R}^{\geq
0}$ is continuous in both $x$ and $\mathcal{L}$ variables, which implies that
$\ell_{\mathcal{L}}(r(t))\leq 1$ for any $t<T$. Thus

(8.1) $\lim_{t\to T}\ell_{\mathcal{L}}(r(t))\leq 1.$

Let $\sigma$ be the maximal multi-curve such that
$r(T)\in\mathcal{S}(\sigma)$. Then for each simple closed curve
$\gamma\in\sigma$ we have that $i(\gamma,\mathcal{L})=0$. Otherwise, since
$\ell_{\gamma}(r(t))\to 0$ as $t\to T$, we would have that
$\ell_{\mathcal{L}}(r(t))\to\infty$ as $t\to T$, which contradicts the bound
(8.1). This as we saw in the previous paragraph implies that $\sigma$ is
disjoint from all $Z_{a}$’s, and thus
$\sigma\subseteq\bigcup_{a=1}^{m}\partial{Z}_{a}$

######  Claim 8.2.

$d_{\operatorname{WP}}(r(T),c_{n})\to\infty$ as $n\to\infty$.

Let $1\leq a\leq m$. By the choice of $Q_{n}$’s $[Q_{n}]\to[\mathcal{L}_{a}]$
as $n\to\infty$ and $[\mathcal{L}_{a}]$ is in the Grormov boundary of
$\mathcal{C}(Z_{a})$ so $d_{Z_{a}}(Q_{n},Q(r(T)))\to\infty$ as $n\to\infty$.
So by the distance formula (2.2) $d(Q(c_{n}),Q(r(T)))\to\infty$ as
$n\to\infty$. Then by Theorem 3.3 (Quasi-Isometric Model)
$d_{\operatorname{WP}}(r(T),c_{n})\to\infty$ as $n\to\infty$.

######  Claim 8.3.

There is $d>0$ such that
$d_{\operatorname{WP}}(c_{n},\overline{\mathcal{S}(\sigma)})$ for all $n$
sufficiently large.

Let $P_{n}$ be the pants decomposition consisting of the curves in
$\pi_{S\backslash\sigma}(Q_{n})$ and $\sigma$. First we show that
$d(Q_{n},P_{n})$ is uniformly bounded. Let $W\subseteq S$ be a non-annular
subsurface. Either $W\pitchfork S\backslash\sigma$ or $W\pitchfork\sigma$.
Suppose that $W\pitchfork S\backslash\sigma$, then let
$\alpha\in\pi_{S\backslash\sigma}(Q_{n})$ be such that $\alpha\pitchfork W$.
Then since $\alpha$ is in both $P_{n}$ and $Q_{n}$ by the triangle inequality
we have $d_{W}(P_{n},Q_{n})\leq d_{W}(P_{n},\alpha)+d_{W}(\alpha,Q_{n})\leq
2$. If not, then $W\pitchfork\sigma$. As we saw above
$\sigma\subseteq\bigcup_{a=1}^{m}\partial{Z}_{a}$ and by the choice of
$Q_{n}$’s for $n$ sufficiently large $\bigcup_{a=1}^{m}\partial{Z}_{a}\subset
Q_{n}$, so $\sigma\subseteq Q_{n}$. Thus again by the triangle inequality
$d_{W}(P_{n},Q_{n})\leq
d_{W}(P_{n},\sigma)+d_{W}(\sigma,Q_{n})+\operatorname{diam}_{W}(\sigma)\leq
3$. Thus $d_{W}(P_{n},Q_{n})\leq 3$ for every $W\subseteq S$. Let $C$ be the
additive constant corresponding to the threshold constant $3$ in the distance
formula (2.2). Then $d(P_{n},Q_{n})\leq C$ for all $n$ sufficiently large.

Let $x_{P_{n}}\in\operatorname{Teich}(S)$ be a point with a Bers pants
decomposition $P_{n}$. By the bound from the previous paragraph Theorem 3.3
implies that $d_{\operatorname{WP}}(c_{n},x_{P_{n}})\leq
K_{\operatorname{WP}}C+C_{\operatorname{WP}}$. Furthermore, $\sigma\subset
P_{n}$ so by Proposition 3.6,
$d_{\operatorname{WP}}(x_{P_{n}},\mathcal{S}(\sigma))\leq\sqrt{2\pi\sum_{\gamma\in\sigma}\ell_{\gamma}}\leq\sqrt{2\pi\xi(S)L_{S}}$.
Then by the triangle inequality
$d_{\operatorname{WP}}(c_{n},\mathcal{S}(\sigma))\leq d$, where $d$ is the sum
of the two upper bounds.

######  Claim 8.4.

For any $t^{\prime}>T$ the points $r_{n}(t^{\prime})$ converge into
$\overline{\mathcal{S}(\sigma)}$.

Let $c^{\prime}_{n}$ be the nearest point to $c_{n}$ on the $\sigma-$stratum.
Let $\eta_{n}$ be the parametrization of $[r(T),c_{n}]$ by arc-length with
$\eta_{n}(0)=r(T)$ and $\eta^{\prime}_{n}$ be the parametrization of
$[r(T),c^{\prime}_{n}]$ by arc-length with $\eta^{\prime}_{n}(0)=r(T)$. Since
$\mathcal{S}(\sigma)$ is geodesically convex
$\eta^{\prime}_{n}\subset\mathcal{S}(\sigma)$. By Claim 8.2
$d_{\operatorname{WP}}(r(T),c_{n})\to\infty$ as $n\to\infty$. By Claim 8.3,
$d_{\operatorname{WP}}(c_{n},\mathcal{S}(\sigma))\leq d$. Then the
$\operatorname{CAT}(0)$ comparison for the triangles with vertices
$r(T),c_{n}$ and $c^{\prime}_{n}$ implies that given any $s$,
$\eta_{n}(s)\to\eta^{\prime}_{n}(s)$ as $n\to\infty$.

Further $\eta_{n}$ and $[r_{n}(T),c_{n}]$ have the same end point $c_{n}$, and
$r_{n}(T)$ converges to $r(T)$ as $n\to\infty$. Then the
$\operatorname{CAT}(0)$ comparison for the triangle with vertices
$r(T),r_{n}(T)$ and $c_{n}$ implies that the Hausdorff distance between
$\eta_{n}$ and $[r_{n}(T),c_{n}]$ tends zero as $n\to\infty$. This together
with the previous paragraph imply that $r_{n}(t^{\prime})$ converges into the
$\sigma-$stratum.

By Claim 8.4 the points $r_{n}(t^{\prime})$ converge to a point
$y\in\overline{\mathcal{S}(\sigma)}$. By the Non-refraction Theorem the
interior of $[x,y]$ is inside the maximal stratum containing its endpoints
which is the Teichmüller space. By the $\operatorname{CAT}(0)$ comparison the
Hausdorff distance of $r|_{[0,t^{\prime}]}$ and $[x,y]$ goes to $0$ as
$n\to\infty$. So $r_{n}(T)$ converges to a point inside the Teichmüller space.
But this contradicts the assumption that $r_{n}(T)$ converge to a point in the
$\sigma-$stratum.

This finishes the proof of that the geodesic $r$ is an infinite geodesic ray.

We proceed to show that the forward ending lamination of $r$ contains
$\nu_{a}$ for $a=1,...,m$. Fix an $a$. Let $t_{n}\in[0,\infty)$ be a sequence
of times such that $d_{\operatorname{WP}}(r(t_{n}),c_{n})\leq d$. Let
$D=K_{\operatorname{WP}}d+C_{\operatorname{WP}}$, then
$d(Q(r(t_{n})),Q(c_{n}))\leq D$. Let $\alpha_{n}\in Q(r(t_{n}))$ be such that
$\alpha_{n}\pitchfork Z_{a}$, then by (6.6)
$d_{Z_{a}}(\alpha_{n},\gamma_{n})\leq D$ (*). Furthermore, $[\gamma_{n}]$
converge to the projective class of measured lamination $\mathcal{E}$ with
support $\nu_{a}$ in the Gromov boundary of $\mathcal{C}(Z_{a})$. Then the
bound (*) implies that after possibly passing to a subsequence $[\alpha_{n}]$
converges to $[\mathcal{E}]$. This follows for example from the definition of
convergence of sequences in the union of a $\delta-$hyperbolic space and its
boundary using the Gromov inner product, see $\S$III.H.3 of [BH99]. Now
$\alpha_{n}$ is a sequence of distinct Bers curves along $r$ so by the
definition of forward ending lamination, $\nu_{a}$ is contained in the forward
ending lamination of $r$.

To prove the last statement of the lemma fix $1\leq a\leq m$. Let
$\alpha\in\partial{Z}_{a}$. For $n$ sufficiently large
$\partial{Z}_{a}\subseteq Q_{n}$, so $\alpha$ is a pinching curve of $r_{n}$.
Furthermore $r_{n}$’s converge to $r$. Then Lemma 3.14 implies that the length
of $\alpha$ is decreasing along $r$.

∎

Let $(\nu^{-},\nu^{+})$ be a narrow pair, where $Z=S\backslash\\{\text{closed
leaves of $\nu^{+}$}\\}$ is a large subsurface and
$\nu^{\prime}=\nu^{+}\backslash\\{\text{closed leaves of $\nu^{+}$}\\}$ is a
minimal filling lamination on $Z$. Let $\rho$ be any infinite hierarchy path
between $\nu^{-}$ and $\nu^{+}$ as above. Recall that by Theorem 5.5 any two
such resolution paths fellow travel each other in the pants graph. The above
lemma provides us with an infinite ray, denoted the ray by $r_{\nu^{\pm}}$,
with forward ending lamination containing $\nu^{\prime}$. The following
theorem shows that the behavior of $r_{\nu^{\pm}}$ mimics the combinatorial
properties of hierarchy paths encoded in the end invariant listed in Theorem
2.13.

######  Theorem 8.5.

(Infinite ray with prescribed itinerary) Given $A,R,R^{\prime}>0$. Let
$(\nu^{-},\nu^{+})$ be an $A-$narrow pair. Let $\rho$ be a hierarchy path
between $\nu^{-}$ and $\nu^{+}$. Let
$r_{\nu^{\pm}}:[0,\infty)\to\operatorname{Teich}(S)$ be the corresponding
infinite WP geodesic ray.

Let $\bar{\epsilon}=\bar{\epsilon}(A,R)$ be the constant from Lemma 6.6 and
for an $\epsilon\leq\bar{\epsilon}$ let
$\bar{w}=\bar{w}(A,R,R^{\prime},\epsilon)$ be the constant form Theorem 6.1.
Assume that $Z$ a large component domain of $\rho$ has
$(R,R^{\prime})-$bounded combinatorics over an interval
$[m^{\prime},n^{\prime}]\subset J_{Z}$ with $n^{\prime}-m^{\prime}>2\bar{w}$.
Then

  1. (1)

$\ell_{\gamma}(r_{\nu^{\pm}}(t))>\bar{\epsilon}$ for every
$\gamma\notin\partial{Z}$, and

  2. (2)

$\ell_{\alpha}(r_{\nu^{\pm}}(t))\leq\epsilon$ for every $\alpha\in\partial{Z}$

for every $t\in[a^{\prime},b^{\prime}]$, where $a^{\prime}\in
N(m^{\prime}+\bar{w})$ and $b^{\prime}\in N(n^{\prime}-\bar{w})$. Here
$N:=N_{\rho,g}$ is the parameter map from Proposition 5.25.

Moreover, if $Z_{1}$ and $Z_{2}$ are subsurfaces as above,
$n^{\prime}_{1}<m^{\prime}_{2}$ implies that $b^{\prime}_{1}<a^{\prime}_{2}$.

###### Proof.

By the narrow assumption given any subsurface $Y\subseteq S$ which is not
large, $d_{Y}(\nu^{-},\nu^{+})\leq A$. So for any $n$ Theorem 2.13 (6) (no
backtracking) implies that $d_{Y}(\rho(0),\rho(n))\leq A+2M_{2}$. Thus the end
invariant of the geodesic segment $[x,c_{n}]$ is $A+2M_{2}$ narrow. Since $Z$
has $(R,R^{\prime})-$bounded combinatorics over $[m,n]$, by Lemma 6.6, there
are $\bar{\epsilon}$ and $w$ such that for any $\gamma\notin\partial{Z}$,
$\ell_{\gamma}(r_{n}(t))$ for every $t\in N(j)$ where
$j\in[m^{\prime}+w,n^{\prime}-w]$. Moreover, since $w\leq\bar{w}$ the bounds
hold on the interval $[a^{\prime},b^{\prime}]$. Furthermore, by Theorem 6.1,
for every $\alpha\in\partial{Z}$, $\ell_{\alpha}(r_{n}(t))\leq\epsilon$ for
every $t\in[a^{\prime},b^{\prime}]$. Since the geodesic rays $r_{n}$ converge
to $r_{\nu^{\pm}}$ point-wise, by Theorem 3.1(Continuity of length-functions)
on Teichmüller space, the same bounds on length-functions hold along
$r_{\nu^{\pm}}$. Finally, the statement about order of intervals follows from
fellow traveling. ∎

We will refer to this ray as a ray with prescribed itinerary.

###  8.2. Divergent geodesic rays

In this subsection we construct divergent Weil-Petersson geodesics in the
moduli space. A ray is divergent if eventually leaves every compact subset of
moduli space. The existence of uncountably many divergent rays starting at a
given point with minimal filling ending lamination is a consequence of our
construction in this subsection.

######  Definition 8.6.

A geodesic ray $r:[0,\infty)\to\mathcal{M}(S)$ is recurrent to a compact
subset $\mathcal{K}\subset\mathcal{M}(S)$, if there is a sequence of times
$t_{i}\to\infty$ as $i\to\infty$ such that $r(t_{i})\in\mathcal{K}$. A ray
$r:[0,\infty)\to\mathcal{M}(S)$ is divergent if it is not recurrent to any
compact subset of moduli space. In other words, $r$ is divergent if for every
compact set $\mathcal{K}\subset\mathcal{M}(S)$, there is a $T\geq 0$ such that
$r([T,\infty))$ does not intersect $\mathcal{K}$.

###### Proof of Theorem 1.2.

Let the indexed subsurfaces $X_{0},X_{1},X_{2}$ and $X_{3}$, and partial
pseudo-Anosov maps $f_{0},f_{1},f_{2}$ and $f_{3}$ supported on them
respectively be as in $\S$ 7.1. Recall that $X_{0}$ and $X_{2}$ are the same
subsurface with different indices. Let the function $q=q_{0}$ where
$q_{0}(i)\equiv i$ (mod 4) and the sequence of integers $e_{i}>E$ be as in
Proposition 7.12. For each $i\geq 1$ let

$Z_{i}=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}.$

Fix a marking $\mu_{I}$ containing $\\{\partial{X}\\}_{a=0,1,2,3}$ and let
$\mu_{T}$ be as in Proposition 7.12. Then by part (i) of the proposition there
are constants $K_{1},C_{1}$ and $E>0$ such that if $|e_{i}|>E$ then

(8.2) $d_{Z_{i}}(\mu_{I},\mu_{T})\asymp_{K_{1},C_{1}}|e_{i}|.$

In particular, by the choice of $E$,

$d_{Z_{i}}(\mu_{I},\mu_{T})>4M.$

By Proposition 7.12 (ii) every subsurface $Z$ with
$d_{Z}(\mu_{I}(q,e),\mu_{T}(q,e))>\textbf{m}$ is in the list of the
subsurfaces $Z_{i}$ and consequently is a large subsurface. So the pair
$(\mu_{I},\mu_{T})$ is $\textbf{m}-$narrow. Let $\rho:[0,\infty]\to P(S)$ be a
hierarchy path between $\mu_{I}$ and $\mu_{T}$. Let
$r:[0,\infty)\to\operatorname{Teich}(S)$ be the WP geodesic ray with end
invariant $(\mu_{I},\mu_{T})$ and prescribed itinerary as in Theorem 8.5. By
Proposition 5.25 we have the parameter map $N$ from the parameters of $\rho$
to the parameters of $r$.

For each $i$ odd, let $k_{i}=\max J_{Z_{i}}\cap J_{Z_{i-1}}$ and $l_{i}=\min
J_{Z_{i}}\cap J_{Z_{i+1}}$ and suppose that $l_{i}>k_{i}$. For each $i$ even,
let $k_{i}=\min J_{Z_{i}}$ and $l_{i}=\max J_{Z_{i}}$. Note that when $i$ is
even $J_{Z_{i}}=[k_{i},l_{i}]$.

First we collect some subsurface coefficient bounds.

$i$ is even. Let $J_{Z_{i}}=[j^{-},j^{+}]$ (when $i$ is even $k_{i}=j_{i}^{-}$
and $l_{i}=j_{i}^{+}$). By Proposition 7.12 (ii) and no backtracking (6.1) for
every subsurface which is not in the list of $Z_{i}$’s we have that

(8.3) $d_{W}(\rho(j^{-}),\rho(j^{+}))\leq\textbf{m}+2M.$

By Proposition 7.12 (iv), for all $j\geq 1$ with $j-i\geq 2$, $Z_{i}<Z_{j}$
and for all $j$ with $i-j\geq 2$, $Z_{j}<Z_{i}$. Suppose that $Z_{i}<Z_{j}$.
Then by Proposition 2.15 $d_{Z_{j}}(\mu_{I},\partial{Z}_{i})\leq M$.
Furthermore $\partial{Z}_{i}\subset\rho(j^{-})$ and
$\partial{Z}_{i}\subset\rho(j^{+})$. So $d_{Z_{j}}(\mu_{I},\rho(j^{-}))\leq
M+\operatorname{diam}_{Z_{j}}(\rho(j^{-}))=M+1$ and
$d_{Z_{j}}(\mu_{I},\rho(j^{+}))\leq M+1$. Then by the triangle inequality

$d_{Z_{j}}(\rho(j^{-}),\rho(j^{+}))\leq
d_{Z_{j}}(\mu_{I},\rho(j^{+}))+d_{Z_{j}}(\mu_{I},\rho(j^{-}))+4\leq 2M+4.$

If $Z_{j}<Z_{i}$, similarly we can get the same bound. We recored these bounds

(8.4) $d_{Z_{j}}(\rho(j^{-}),\rho(j^{+}))\leq 2M+4.$

$i$ is odd. Let $J_{Z_{i}}=[j^{-},j^{+}]$. By Proposition 7.12 (iii) and no
backtracking (6.1) for every subsurface $W$ which is not in the list of
$Z_{i}$’s,

(8.5) $d_{W}(\rho(j^{-}),\rho(j^{+}))\leq\textbf{m}+2M.$

and

(8.6) $d_{W}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}+2M.$

By Proposition 7.12 (iv), for all $j\geq 1$ with $j-i\geq 2$, $Z_{i}<Z_{j}$
and for all $j$ with $i-j\geq 2$, $Z_{j}<Z_{i}$. Then similar to the proof of
(8.4) we can get

(8.7) $d_{Z_{j}}(\rho(j^{-}),\rho(j^{+}))\leq 2M+4,$

and

(8.8) $d_{Z_{j}}(\rho(k_{i}),\rho(l_{i}))\leq 2M+4.$

By Theorem 2.13 (6)

$d_{Z_{i-1}}(\rho(k_{i}),\rho(l_{i}))+d_{Z_{i-1}}(\rho(l_{i}),\mu_{T})\leq
d_{Z_{i-1}}(\rho(k_{i}),\mu_{T})+M.$

Now by Theorem 2.13 (4), $d_{Z_{i-1}}(\mu_{T},\rho(k_{i}))\leq M$. Then by the
above inequality

(8.9) $d_{Z_{i-1}}(\rho(k_{i}),\rho(l_{i}))\leq 2M.$

Similarly,

$d_{Z_{i+1}}(\rho(k_{i}),\mu_{I})+d_{Z_{i+1}}(\rho(k_{i}),\rho(l_{i}))\leq
d_{Z_{i+1}}(\rho(l_{i}),\mu_{I})+M,$

then since $d_{Z_{i+1}}(\mu_{I},\rho(l_{i}))\leq M$ we get

(8.10) $d_{Z_{i+1}}(\rho(k_{i}),\rho(l_{i}))\leq 2M.$

Finally, for any $i$ by Proposition 7.12 (iii) and no backtracking for any
$\gamma\in\mathcal{C}_{0}(S)$,

(8.11) $d_{\gamma}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}^{\prime}+2M.$

Now we proceed to estimate the length of $J$ intervals (see Theorem 2.13 (1))
using the above subsurface coefficient bounds.

For each $i$ even by the bounds (8.3) and (8.4) all of the subsurface
coefficients of $\rho(j^{-})$ and $\rho(j^{+})$ except that of $Z_{i}$ are
bounded above by $\textbf{m}+2M$. Let the threshold constant in the distance
formula (2.2) be $\textbf{m}+2M$ (note that it is larger that $M_{1}$) and let
$K_{2},C_{2}$ be the constants corresponding to this threshold constant. Then
we have that

$d(\rho(j^{-}),\rho(j^{+}))\asymp_{K_{2},C_{2}}d_{Z_{i}}(\rho(j^{-}),\rho(j^{+})).$

Similarly for each $i$ odd by the bounds (8.5) and (8.7) and the distance
formula we have that

$d(\rho(j^{-}),\rho(j^{+})))\asymp_{K_{2},C_{2}}d_{Z_{i-1}}(\rho(j^{-}),\rho(j^{+}))+d_{Z_{i}}(\rho(j^{-}),\rho(j^{+}))+d_{Z_{i+1}}(\rho(j^{-}),\rho(j^{+})).$

By the no backtracking (6.1) and Theorem 2.13 (4) each subsurface coefficient
on the right hand side is $(1,2M)$ comparable with the corresponding
subsurface coefficient of $\mu_{I}$ and $\mu_{T}$. For example
$d_{Z_{i-1}}(\rho(j^{-}),\rho(j^{+}))\asymp_{1,2M}d_{Z_{i-1}}(\mu_{I},\mu_{T})$.
Moreover $\rho$ is a $(k,c)-$quasi-geodesic where $k,c$ depend only on the
topological type of $S$. Let $K_{3}=kK_{2}$ and $C_{3}=k(C_{2}+2M)+c$. Then we
have:

For each $i$ even

(8.12) $|J_{Z_{i}}|\asymp_{K_{3},C_{3}}d_{Z_{i}}(\mu_{I},\mu_{T})$

and for each $i$ odd,

(8.13)
$|J_{Z_{i}}|\asymp_{K_{3},C_{3}}d_{Z_{i-1}}(\mu_{I},\mu_{T})+d_{Z_{i}}(\mu_{I},\mu_{T})+d_{Z_{i+1}}(\mu_{I},\mu_{T})$

We proceed to set the sequence $\\{e_{i}\\}_{i}$ and using the above estimates
finish the construction of divergent WP geodesic rays.

Let $\epsilon_{j}\to 0$ be a decreasing sequence. Let $R=\textbf{m}+2M$ and
$R^{\prime}=\textbf{m}^{\prime}+2M$. For each $j\geq 1$ let

(8.14) $y_{j}\geq 2\bar{w}(\textbf{m},R,R^{\prime},\epsilon_{j})$

where $\bar{w}$ is the constant from Theorem 8.5.

Let $K=K_{3}K_{1}$ and $C=K_{1}C_{1}+K_{1}K_{2}C_{3}$. Define the sequence
$\\{e_{i}\\}_{i}$ as follows: For $i$ even let $e_{i}=Ky_{\frac{i}{2}}+KC$,
and for $i$ odd let
$e_{i}=(K^{3}y_{\frac{i-1}{2}}+K^{2}C)+(K^{3}y_{\frac{i+1}{2}}+K^{3}C)$.

We investigate the pattern that the $J$ intervals overlap and estimate the
length of each interval $[k_{i},l_{i}]$.

$i$ is even. Recall that when $i$ is an even integer
$[k_{i},l_{i}]=J_{Z_{i}}$. Then (8.12) implies that
$l_{i}-k_{i}\geq\frac{1}{K_{3}}d_{Z_{i}}(\mu_{I},\mu_{T})-C_{3}$. Then by
(8.2) $l_{i}-k_{i}\geq\frac{1}{K_{3}K_{1}}|e_{i}|-\frac{C_{1}}{K_{3}}-C_{3}$.
Therefore, $l_{i}-k_{i}\geq y_{\frac{i}{2}}$.

$i$ is odd. By (8.12) and (8.2) $|J_{Z_{i-1}}|\leq K|e_{i-1}|+C$, so
$|J_{Z_{i}}\cap J_{Z_{i-1}}|\leq K|e_{i-1}|+C$. Similarly, $|J_{Z_{i+1}}|\leq
K|e_{i+1}|+C$ and $|J_{Z_{i}}\cap J_{Z_{i+1}}|\leq K|e_{i+1}|+C$. Furthermore,
by (8.13) and (8.2),
$|J_{Z_{i}}|\geq\frac{1}{K}(|e_{i-1}|+|e_{i}|+|e_{i+1}|)-C$.

By Proposition 7.12 (v) for every $j\in J_{Z_{i}}$, $j\geq\min J_{Z_{i-1}}$,
so $J_{Z_{i-1}}$ and $J_{Z_{i}}$ intersect as in Figure 10. Furthermore, by
the choice of $e_{i}$’s and the bounds above $|J_{Z_{i}}|>|J_{Z_{i}}\cap
J_{Z_{i-1}}|$, thus $l_{i-1}<k_{i}$. Similarly by Proposition 7.12 (v) for
every $j\in J_{Z_{i}}$, $j\geq\max J_{Z_{i+1}}$ so $J_{Z_{i+1}}$ and
$J_{Z_{i}}$ intersect as in Figure 10. Furthermore,
$|J_{Z_{i}}|>|J_{Z_{i}}\cap J_{Z_{i-1}}|$ and $l_{i}<k_{i+1}$.

Finally putting together the bounds on the length of $J_{Z_{i}}$ and its
intersection with the intervals $J_{Z_{i-1}}$ and $J_{Z_{i+1}}$ above, we
obtain $|J_{Z_{i}}-(J_{Z_{i-1}}\cup J_{Z_{i+1}})|=|J_{Z_{i}}|-|J_{Z_{i}}\cap
J_{Z_{i-1}}|-|J_{Z_{i}}\cap J_{Z_{i+1}}|\geq y_{\lfloor\frac{i}{2}\rfloor}$.
Therefore, $l_{i}-k_{i}\geq y_{\lfloor\frac{i}{2}\rfloor}$ and $l_{i}>k_{i}$.
See Figure 10.

Let $R=\textbf{m}+2M$ and $R^{\prime}=\textbf{m}^{\prime}+2M$. When $i$ is
even by the subsurface coefficient bounds (8.3), (8.4) and (8.11) $Z_{i}$ has
$(R,R^{\prime})-$bounded combinatorics over the interval $[k_{i},l_{i}]$. When
$i$ is odd by the bounds (8.6), (8.8), (8.9), (8.10) and (8.11) the subsurface
$Z_{i}$ has $(R,R^{\prime})-$bounded combinatorics over the interval
$J_{Z_{i}}=[k_{i},l_{i}]$.

For each $i\geq 1$ let $t_{i}\in N(j)$ for some
$j\in[k_{i}+\bar{w},l_{i}-\bar{w}]$. For any $i\geq 1$, as we saw above,
$l_{i}<k_{i+1}$ so we have that $t_{i}<t_{i+1}$. Theorem 8.5 applied to the
interval $[k_{i},l_{i}]$ implies that

$\ell_{\partial{Z_{i}}}(r(t_{i}))\leq\epsilon_{\lfloor\frac{i}{2}\rfloor},$

($\ell_{\partial{Z}}(x)=\max\\{\ell_{\alpha}(x):\alpha\in\partial{Z}\\}$).

By the definition of $q$ any two consecutive domains $Z_{i}$ and $Z_{i+1}$
have a boundary curve in common
$\delta_{i}:=\partial{Z_{i}}\cap\partial{Z_{i+1}}$. Then by the convexity of
length-functions we have

$\ell_{\delta_{i}}(r(t))\leq\epsilon_{\lfloor\frac{i}{2}\rfloor}\;\text{for
every}\;t\in[t_{i},t_{i+1}]$

Now since the intervals $[t_{i},t_{i+1}]$, $i\geq 1$, cover $[0,\infty)$, the
domain of $r$, and $\epsilon_{i}\to 0$, the systole of the surfaces along the
WP geodesic ray $r$ decreases and goes to 0.

By the Mumford’s compactness criterion every compact subset of moduli space is
contained in some $\epsilon-$thick part. So $\hat{r}$ the projection of $r$ to
the moduli space is a divergent geodesic ray. Finally, by Proposition 7.12
(vi), $\mu_{T}(q,e)$ is a minimal filling lamination, so the forward ending
lamination of $\hat{r}$ is minimal filling.

Note that there are uncountably sequences $\\{y_{j}\\}_{j\geq 1}$ which
satisfy inequality (8.14). Consequently, we have uncountably many divergent
rays starting from a given point in the moduli space. These geodesics are
distinct because have different forward ending lamination. ∎

Figure 10. The intersection pattern of the intervals $J_{Z_{i-1}}$,
$J_{Z_{i}}$ and $J_{Z_{i+1}}$ when $i$ is even. Over the red subintervals of
$J$ intervals the corresponding domain has bounded combinatorics. $k_{i}=\max
J_{Z_{i-1}}\cap J_{Z_{i}}$ and $l_{i}=\min J_{Z_{i+1}}\cap J_{Z_{i}}$.

###  8.3. Closed geodesics in the thin part

In this subsection we provide examples of closed Weil-Petersson geodesics
which stay in the thin part of moduli space.

###### Proof of Theorem 1.1.

Let the indexed subsurfaces $X_{0},X_{1},X_{2}$ and $X_{3}$, and partial
pseudo-Anosov maps $f_{0},f_{1},f_{2}$ and $f_{3}$ supported on them
respectively be as in $\S$ 7.1. Recall that $X_{0}$ and $X_{2}$ are the same
subsurfaces with different indices. Let the function $q=q_{0}$ and the
sequence of integers $e_{i}>E$ ($i\geq 1$) be as in Proposition 7.12. For each
$i\geq 1$ let

$Z_{i}=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}.$

Fix a marking $\mu_{I}$ containing $\\{\partial{X}\\}_{a=0,1,2,3}$ and let
$\mu_{T}$ be as in Proposition 7.12.

Then we are in the set up of the proof of Theorem 1.2 in $\S$8.2. As we saw
there the pair $(\mu_{I},\mu_{T})$ is $\textbf{m}-$narrow where m is the
constant form Proposition 7.12. Let $\rho:[0,\infty]\to P(S)$ be a hierarchy
path between $\mu_{I}$ and $\mu_{T}$. Let
$r:[0,\infty)\to\operatorname{Teich}(S)$ be the WP geodesic ray with end
invariant $(\mu_{I},\mu_{T})$ and prescribed itinerary as in Theorem 8.5. Let
$N$ be the parameter map from Proposition 5.25.

As in the proof of Theorem 1.2 for each $i$ odd let $k_{i}=\max
J_{Z_{i-1}}\cap J_{Z_{i}}$ and $l_{i}=\min J_{Z_{i+1}}\cap J_{Z_{i}}$. For
each $i$ even, let $k_{i}=\min J_{Z_{i}}$ and $l_{i}=\max J_{Z_{i}}$.

Then all of the subsurface coefficient bounds (8.3)-(8.11) in the proof of
Theorem 1.2 hold.

Let $R=\textbf{m}+2M$ and $R^{\prime}=\textbf{m}^{\prime}+2M$. Given
$\epsilon>0$ let

(8.15) $y\geq 2\bar{w}(\textbf{m},R,R^{\prime},\epsilon).$

where $\bar{w}$ is the constant from Theorem 8.5.

Let the constants $K\geq 1$ and $C\geq 0$ be as in the proof of Theorem 1.2.
Define the periodic sequence $e_{i}$ with $e_{i}=e_{i^{\prime}}$ whenever
$i\equiv i^{\prime}$ (mod 4) and the first four terms
$e_{1}=2(K^{3}y+K^{3}C),e_{2}=Ky+KC,e_{3}=2(K^{3}y+K^{3}C),e_{4}=Ky+KC$.

The estimates (8.12) and (8.13) on the length of $J_{Z_{i}}$ intervals hold.
Then similar to the proof of Theorem 1.2 we can show that the $J$ intervals
intersect as in Figure 10, $l_{i}<k_{i+1}$ and $l_{i}-k_{i}\geq y$ for all
$i\geq 1$.

When $i$ is even by the subsurface coefficient bounds (8.3), (8.4) and (8.11)
$Z_{i}$ has $(R,R^{\prime})-$bounded combinatorics over the interval
$[k_{i},l_{i}]$. When $i$ is odd by the bounds (8.6), (8.8), (8.9), (8.10) and
(8.11) the subsurface $Z_{i}$ has $(R,R^{\prime})-$bounded combinatorics over
the interval $[k_{i},l_{i}]$.

Let $t_{i}\in N(j)$ where $j\in[k_{i}+\bar{w},l_{i}-\bar{w}]$. Theorem 8.5
applied to the interval $[k_{i},l_{i}]$ implies that

$\ell_{\partial{Z_{i}}}(r(t_{i}))\leq\epsilon$

Furthermore since $l_{i}<k_{i+1}$, $t_{i+1}>t_{i}$.

By the definition of $q$ any two consecutive domains $Z_{i}$ and $Z_{i+1}$
have a boundary curve in common
$\delta_{i}=\partial{Z_{i}}\cap\partial{Z_{i+1}}$. Then by the convexity of
length-functions we have that

$\ell_{\delta_{i}}(r(t))\leq\epsilon\;\text{for every}\;t\in[t_{i},t_{i+1}]$

Now since the intervals $[t_{i},t_{i+1}]$, $i\geq 1$, cover $[0,\infty)$, at
any time the systole of the surface along $r$ is less than $\epsilon$.
Therefore, $r$ stays in the $\epsilon-$thin part of Teichmüller space.
Consequently, $\hat{r}$ the projection of $r$ to the moduli space stays in the
$\epsilon-$thin part of moduli space. Furthermore, since $q$ and
$\\{e_{i}\\}_{i}$ are periodic $\hat{r}$ is a closed geodesic.

By Mumford’s compactness criterion there is $\epsilon_{0}>0$ such that
$\mathcal{K}$, the compact subset of moduli space, is contained in the
$\epsilon_{0}-$thick part of the moduli space and consequently is disjoint
from the $\epsilon_{0}-$thin part of the moduli space. Let $y$ be such that
(8.15) holds for $\epsilon_{0}$. Then our construction produces closed WP
geodesics not intersecting $\mathcal{K}$. There are infinitely many integers
$y$ which satisfy this condition and consequently there are infinitely many
closed geodesics not intersecting $\mathcal{K}$. These geodesics are distinct
because have different forward ending lamination. ∎

###  8.4. A recurrence condition

Given $A,R$ and $R^{\prime}$ positive. Let the constants $w=w(A,R)$ and
$\bar{\epsilon}=\bar{\epsilon}(A,R,R^{\prime})$ be form Proposition 6.7. The
following theorem is a straightforward consequence of the proposition.

######  Theorem 8.7.

(Recurrence condition) Let $(\mu^{-},\mu^{+})$ be an $A-$narrow pair. Let
$\rho$ be a hierarchy path between $\mu^{-}$ and $\mu^{+}$. Let
$[k_{i},l_{i}]$, $i\geq 1$, be a sequence of intervals with $l_{i}-k_{i}\geq
2w$ and $l_{i}<k_{i+1}$. Furthermore suppose that over each interval
$[k_{i},l_{i}]$, $S$ has $(R,R^{\prime})-$bounded combinatorics. Let $r$ be
the ray with prescribed itinerary with end invariant $(\mu^{-},\mu^{+}$). Then
$\operatorname{inj}r(t_{i})\geq\frac{\bar{\epsilon}}{2}$, where $t_{i}\in
N(j)$ for some $j\in[k_{i}+w,l_{i}-w]$.

The WP volume of the moduli space is finite. It follows for example from the
fact that given $\bar{\epsilon}>0$ the WP metric extends to the Delinge-
Mumford compactification of the moduli space, [Mas76]. So the Poincaré
Recurrence Theorem implies that almost every WP geodesic is recurrent to the
$\frac{\bar{\epsilon}}{2}-$thick part of moduli space. But here our
construction of recurrent WP geodesics to the $\bar{\epsilon}-$thick part of
moduli space only uses the combinatorial control we developed in this paper.

######  Theorem 8.8.

There are WP geodesic rays recurrent to the $\bar{\epsilon}-$thick part of
moduli space.

###### Proof.

Let the subsurfaces $X_{0}=S$ and $X_{1}$ and the pseudo-Anosov maps $f_{0}$
and $f_{1}$ supported on $X_{0}$ and $X_{1}$, respectively be as in $\S$ 7.2.
Let the function $q=q_{0}$ and the sequence of $e_{i}>E$ be as in Proposition
7.14. For each $i\geq 1$ let
$Z_{i}=f_{q(1)}^{e_{1}}...f_{q(i-1)}^{e_{i-1}}X_{q(i)}$. Let $\mu_{I}$ be a
marking whose base contains $\\{\partial{X}_{a}\\}_{a=0,1}$ and $\mu_{T}$ be
as in Proposition 7.14.

By Proposition 7.14 (i) we have that

(8.16) $d_{Z_{i}}(\mu_{I},\mu_{T})\asymp_{K_{1},C_{1}}|e_{i}|.$

By Propositions 7.14 (iii) every subsurface $Z$ with
$d_{Z}(\mu_{I}(q,e),\mu_{T}(q,e))>\textbf{m}$ is a large subsurface. So the
pair $(\mu_{I},\mu_{T})$ is ${\bf m}-$narrow.

Let $r$ be the ray with the end invariant $\mu_{I}$ and $\mu_{T}$. Let $N$ be
the parameter map form Proposition 5.25.

For $i$ even let $k_{i}=\max J_{Z_{i-1}}$ and $l_{i}=\min J_{Z_{i+1}}$. For
$i$ odd, let $k_{i}=\min J_{Z_{i}}$ and $l_{i}=\max J_{Z_{i}}$. Note that when
$i$ is odd $J_{Z_{i}}=[k_{i},l_{i}]$.

We have the following subsurface coefficient bounds:

$i$ is odd: Proposition 7.14 (iii) and (6.1) (no backtracking) imply that for
every non-annular subsurface $W$ which is not in the list of $Z_{i}$’s,

(8.17) $d_{W}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}+2M.$

By Proposition 7.14 (ii) for every odd integer $j>i$, $Z_{i}<Z_{j}$, also for
every odd integer $j<i$, $Z_{j}<Z_{i}$. Then an argument similar to the one we
gave to prove (8.4) in the proof of Proposition 7.14 we obtain the bound

(8.18) $d_{Z_{j}}(\rho(k_{i}),\rho(l_{i}))\leq 2M+4$

$i$ is even: Proposition 7.14 (iii) and (6.1) (no backtracking) imply that for
every non-annular subsurface $W$ which is not in the list of $Z_{i}$’s,

(8.19) $d_{W}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}+2M.$

$k_{i}=\max J_{Z_{i-1}}$ and $l_{i}=\min J_{Z_{i+1}}$, so
$J_{Z_{i-1}}\cap[k_{i},l_{i}]=\emptyset$ and
$J_{Z_{i+1}}\cap[k_{i},l_{i}]=\emptyset$. Then by Theorem 2.13 (4),

(8.20) $\displaystyle d_{Z_{i-1}}(\rho(k_{i}),\rho(l_{i}))\leq
2M,\;\text{and}$ $\displaystyle d_{Z_{i+1}}(\rho(k_{i}),\rho(l_{i}))\leq 2M.$

Further, for any odd $j<i-1$, $Z_{j}<Z_{i}$ and for any odd $j>i+1$,
$Z_{i}<Z_{j}$. Then an argument similar to one for the proof of (8.4) gives us

(8.21) $d_{Z_{j}}(\rho(k_{i}),\rho(l_{i}))\leq 2M+4$

By Propositions 7.14 (iv) and the no backtracking for every annular subsurface
$A(\gamma)$,

(8.22) $d_{\gamma}(\rho(k_{i}),\rho(l_{i}))\leq\textbf{m}^{\prime}+2M.$

We proceed to estimate the length of the intervals $[k_{i},l_{i}]$.

$i$ is odd: By the bounds (8.17) and (8.18) all of the subsurface coefficients
of $\rho(k_{i})$ and $\rho(l_{i})$ except that of $Z_{i}$ are bounded above by
$\textbf{m}+2M$. Let the threshold constant in the distance formula (2.2) be
$\textbf{m}+2M$ and let $K_{2},C_{2}$ be the constants corresponding to this
threshold constant. Then we get

$d(\rho(k_{i}),\rho(l_{i}))\asymp_{K_{2},C_{2}}d_{Z_{i}}(\rho(k_{i}),\rho(l_{i}))$

By the no backtracking
$d_{Z_{i}}(\rho(k_{i}),\rho(l_{i}))\asymp_{1,2M}d_{Z_{i}}(\mu_{I},\mu_{T})$.
Furthermore, $\rho$ is a $(k,c)-$quasi-geodesic where $k$ and $c$ only depend
on the topological type of the surface. Let
$K_{3}=kK_{2},C_{3}=k(C_{2}+2M)+c$. Then

(8.23) $l_{i}-k_{i}\asymp_{K_{3},C_{3}}d_{Z_{i}}(\mu_{I},\mu_{T}).$

$i$ is even: By the bound (7.16) in the proof of Proposition 7.14 we have that

(8.24) $d_{S}(\rho(k_{i}),\rho(l_{i}))\geq\frac{1}{K^{\prime}_{1}}|e_{i}|.$

where as in the proposition $K^{\prime}_{1}=\min\\{\tau_{a}:a=0,1\\}$.

By the bounds (8.19), (8.20) and (8.21) all of the subsurface coefficients of
the pair $\rho(k_{i})$ and $\rho(l_{i})$ are bounded above by $\textbf{m}+2M$.
Let $K_{2},C_{2}$ be the constants corresponding to the threshold constant
$\textbf{m}+2M$ in the distance formula (2.2). Then we have

$d(\rho(k_{i}),\rho(l_{i}))\asymp_{K_{2},C_{2}}d_{S}(\rho(k_{i}),\rho(l_{i})).$

Note that $\rho$ is a $(k,c)-$quasi-geodesic. Let $K^{\prime}_{3}=kK_{2}$ and
$C^{\prime}_{3}=kC_{2}+c$. Then we have

(8.25)
$l_{i}-k_{i}\asymp_{K^{\prime}_{3},C^{\prime}_{3}}d_{S}(\rho(k_{i}),\rho(l_{i})).$

We proceed to set the sequence $\\{e_{i}\\}_{i}$ and using the above estimates
finish the construction of recurrent WP geodesic rays.

Let $R=\textbf{m}+2M$ and $R^{\prime}=\textbf{m}^{\prime}+2M$. For any $i$
even let

$y_{i}\geq 2w(\textbf{m},R)$

Let $K^{\prime}=K^{\prime}_{1}K^{\prime}_{3}$ and
$C^{\prime}=K^{\prime}_{1}K^{\prime}_{3}C^{\prime}_{1}$. Define the sequence
$\\{e_{i}\\}_{i}$ such that for any $i$ even
$e_{i}=K^{\prime}y_{i}+C^{\prime}$. For the next paragraph $e_{i}$ could be
any integer when $i$ is odd.

Then the bounds (8.24) and (8.25) imply that $l_{i}-k_{i}\geq 2w$.

Furthermore, by the bounds (8.19), (8.18) and (8.22) $S$ has
$(R,R^{\prime})-$bounded combinatorics over the interval $[k_{i},l_{i}]$. Let
$t_{i}\in N(j)$ where $j\in[k_{i},l_{i}]$. Applying Theorem 8.7 (Recurrence
condition) the geodesic ray $r$ with end invariant $\mu_{I}$ and $\mu_{T}$ has
injectivity radius at least
$\frac{1}{2}\bar{\epsilon}=\frac{1}{2}\bar{\epsilon}({\bf m},R,R^{\prime})$ at
each $t_{i}$. Consequently, $\hat{r}$ the projection of $r$ to the moduli
space is recurrent the $\frac{\bar{\epsilon}}{2}-$thick part of the moduli
space.

Let $R=\textbf{m}+2M$ and $R^{\prime}=\textbf{m}^{\prime}+2M$, as before. Let
$\epsilon_{j}\to 0$ as $j\to\infty$. For each $i$ odd let

$y_{i}\geq
2\bar{w}(\textbf{m},R,R^{\prime},\epsilon_{\lfloor\frac{i}{2}\rfloor})$

Let $K=K_{1}K_{3}$, $C=K_{1}C_{1}+K_{1}K_{3}C_{3}$. Now in the sequence
$\\{e_{i}\\}_{i}$ we set above let for each $i$ odd, $e_{i}\geq Ky_{i}+C$.

Then the bounds (8.23) and (8.16) imply that $l_{i}-k_{i}>2\bar{w}$.

Furthermore, by the bounds (8.17), (8.20), (8.21) and (8.22), $Z_{i}$ has
$(R,R^{\prime})-$bounded combinatorics over the interval $[k_{i},l_{i}]$. Let
$t_{i}\in N(j)$ where $j\in[k_{i}+\bar{w},l_{i}-\bar{w}]$. Then Theorem 8.5
applied to the interval $[k_{i},l_{i}]$ implies that
$\ell_{\partial{Z}_{i}}(r(t_{i}))\leq\epsilon_{\lfloor\frac{i}{2}\rfloor}$.
Thus since $\epsilon_{j}\to 0$ as $j\to\infty$ the ray $\hat{r}$ is not
contained in any thick part of the moduli space. As a result $\hat{r}$ is
recurrent to the $\frac{\bar{\epsilon}}{2}-$thick part of the moduli space and
is not contained in any compact subset of the moduli space.

∎

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# A New Guest in the Third Quantized Multiverse

Samuel Barroso Bellido<EMAIL_ADDRESS>Institute of
Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
Fabian Wagner<EMAIL_ADDRESS>Institute of Physics, University of
Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland

###### Abstract

In the present article we point out a conceptual issue of the Third
Quantization formalism for Canonical Quantum Gravity. Based on earlier results
on interuniversal entanglement, the picture of noninteracting universe-
antiuniverse pairs is incomplete. In particular, the variation of the
entanglement entropy with the scale factor, understood as a measure of time,
implies either an interaction between pairs of universes or universes with a
heat bath, i. e. an external field. We propose a simple solution to the
problem by adding a new quantum field to the formalism. Correspondingly,
particles of this field mediate interactions between universes – depending on
the model at tree level or one loop – and may further provide a heat bath in
the background.

##  I The Problem of Third Quantization of Canonical Quantum Gravity

Canonical Quantum Gravity [1, 2] is directed at a description of the whole
universe by means of a universal wave function
$\Psi(g_{\mu\nu},\text{matter})$[3, 4]. This wave function of the universe is
considered to be a complex scalar in most of the literature except for some
works in which the authors endow it with spinorial properties (see e.g. [5])
or define it over different fields (in the sense of Mathematics) like
quarternions [6]. Hence, as usual, we will consider it to be a complex scalar
function.

Fundamentally, the wave function of the universe, living in the infinite-
dimensional Hilbert space, the so-called superspace, satisfies the so-called
Wheeler-DeWitt (WDW) equation whose general solution, however, is unknown.
Therefore, by analogy with the case of classical General Relativity, it is
useful to reduce the complexity of the problem by imposing symmetries. If we,
for example, consider a FLRW universe with one degree of freedom representing
the matter content, specifically a homogeneous scalar field $\phi$, the
superspace is reduced to a two-dimensional space, the minisuperspace, whose
variables are the said scalar field $\phi$ and the scale factor $a$,
indicative for the expansion of the universe. For a more practical purpose, it
is convenient to use a parameterization of the scale factor as
$\alpha=\ln(a)$. In this case the WDW equation can be expressed as [2, 7]

$\left[\partialderivative[2]{\alpha}-\partialderivative[2]{\phi}+m^{2}(\alpha,\phi)\right]\Psi(\alpha,\phi)=0,$
(1)

where the third term reads

$m^{2}(\alpha,\phi)=\text{e}^{6\alpha}\left[\frac{\Lambda}{3}+2V(\phi)\right]-\text{e}^{4\alpha}K,$
(2)

in a rather general FLRW universe with curvature index $K$, a scalar field
$\phi$ with potential $V(\phi)$ and a cosmological constant $\Lambda$, and the
factor ordering has been chosen such that the momentum associated to the scale
factor becomes

$p_{a}^{2}=-\frac{1}{a}\partialderivative{a}\left(a\partialderivative{a}\right).$
(3)

Hence, the kinetic part of Eq. (1) can be written as the covariant
d’Alembertian $\partial_{\mu}(\sqrt{g}g^{\mu\nu}\partial_{\nu})/\sqrt{g}$ on a
two-dimensional Minkowski background. Since Eq. (1) is very similar to a
Klein-Gordon (KG) equation, given the hyperbolic nature of the differential
operator, it is customary, but not necessary, to recognize $\alpha$ as a time
variable, $\phi$ as a spatial variable and $m^{2}(\alpha,\phi)$ as a mass
term. Some of the downsides of the resulting theory are discussed e. g. in
Refs. [8, 9] and will not be explicitly dealt with in the present work.

The formalism of Third Quantization [10, 11, 12, 13] goes one step beyond in
an attempt to construct a field theory describing the wave function of the
universe $\Psi(\alpha,\phi)$ as a complex scalar field and the WDW equation
(1) as its equation of motion obtained through varying with respect to
$\Psi^{\dagger}$

$\partialderivative{\alpha}\left(\frac{\delta\mathcal{L}}{\delta(\partial_{\alpha}\Psi^{\dagger})}\right)+\partialderivative{\phi}\left(\frac{\delta\mathcal{L}}{\delta(\partial_{\phi}\Psi^{\dagger})}\right)-\frac{\delta\mathcal{L}}{\delta\Psi^{\dagger}}=0,$
(4)

derived from the third quantized action

$\displaystyle
S=\int\text{d}\alpha\text{d}\phi\left[-\Psi^{\dagger}\square\Psi+m^{2}(\alpha,\phi)\Psi^{\dagger}\Psi\right],$
(5)

where $\square=-\partial_{\alpha}^{2}+\partial_{\phi}^{2}$. Accordingly, the
field $\Psi(\alpha,\phi)$ is treated as the representation of a universe-like
particle, while $\Psi^{\dagger}$ yields the corresponding antiparticle. As a
result, the multiverse is the playground of a bunch of particles
$\Psi(\alpha,\phi)$ and $\Psi^{\dagger}(\alpha,\phi)$ interacting with each
other, exclusively. Annihilation and creation operators have to be defined for
each universe and antiuniverse in order to simplify the treatment, selecting
also the proper Fock space as in Quantum Field Theory.

Based on the scheme presented by Third Quantization, the entanglement entropy
of a pair of universes created as in analogy with pair creation [14, 15] has
been recently calculated for different models in Refs. [16, 7, 17]. It is
important to notice that the number of particles is constant and any vertex
stems from the action (5), so once the pair is born, neither of the two
universes can decay or feel any other kind of interaction. If the multiverse
was just an empty space where universes were freely moving, the entanglement
entropy should not change unless they interact with an environment or another
particle, which none of those cases are considered here. It was found that the
entanglement entropy is not constant, in general [17], for any value of
$\alpha$ and $\phi$, and therefore there should be a source of its
variability.

It may be argued that the entanglement entropy is not relevant but the
production of entropy of the system [2, 18]

$\sigma=\derivative{S}{a}-\frac{1}{T}\frac{\delta Q}{\text{d}a},$ (6)

which considers it as a composite system evolving adiabatically. However, it
does not stand up to the comparison with QED. Universes must be interpreted as
point particles travelling through the vacuum and as such lack internal
structure. Therefore, the net production of entropy is trivially zero, while
the entanglement entropy remains nonvanishing in this picture.

Where does this entanglement of the pair thus come from? They are complex
scalar particles created at the same point on the minisuperspace. Described by
the natural Hamiltonian, usually called diagonal Hamiltonian since is
separable as $H=H_{1}+H_{2}$, particles are not entangled. However, this
picture looks unnatural since the number of universes apparently varies as
time goes by [19, 20, 21, 22], i. e. as the universes evolve. This problem can
be sorted out by choosing a certain representation in which the number of
universes, or equivalently the universal number operator, remains constant,
called the invariant or Lewis-Riesenfeld representation [23, 24]. Then, the
quantum evolution of a pair is stationary, and so is the property of it being
a pair. The comparison with the diagonal representation makes apparent that
the Hamiltonian contains a non-diagonal part which is responsible for the
entanglement [15]. This leads to the immediate conclusion that there is a
wrong interpretation of Third Quantization since it is impossible for a pair
to get a varying entanglement entropy in a non-interacting multiverse.

##  II The Solution: A forgotten guest

Daring a closer look at the reasoning, the analogy with the KG equation was
drawn in a rather naïve way. The Lagrangian of a field theory does not contain
a function $m^{2}(\alpha,\phi),$ which is not described by a field, or the
vacuum expectation value of a field which indeed would be constant, as it
happens with the Higgs mechanism [25, 26]. In other words, there is no
isomorphism between the KG equation and the WDW equation (1) since the masses
of both of them are essentially different. One is a constant, making the
theory analytical manageable, and the other is a function of the internal
degrees of freedom of the universes. Therefore, considering the scale factor
as a time variable, the mass term changes in time. Any dynamics in the
direction of the resulting arrow of time should thus be considered with due
care. Regarding the pair of particles, what it is shocking is the variability
of the entanglement entropy in time. Besides, it is interesting to remark that
the action (5) is not bounded from below when $m^{2}(\alpha,\phi)$ in Eq. (2)
is negative. Yet, this argument may not be as strong as it seems since
$\absolutevalue{\Psi(\alpha,\phi)}^{2}$ is not related to any probabilistic
interpretation as in Quantum Mechanics [27, 28, 29].

In fact, it is more natural to understand the nonconstant mass term as an
interaction with an external field. In that vein and by analogy e. g. with
QED, we may thus introduce a new degree of freedom $\Omega$ accounting for the
dynamics of $m^{2}(\alpha,\phi)$ in the WDW equation (1), which acts as a
source term. Taking a step back by regarding the most general superspace, this
new field would be the cause of the dynamics of the term proportional to [2,
13]

$\sqrt{h}\left[-^{(3)}R+2\Lambda+16\pi\left[K_{\text{s}}+V\right]_{\text{matter}}\right]\supset\mathcal{H},$
(7)

contained into the most general WDW equation, where $\mathcal{H}$ is the
Hamiltonian, ${}^{(3)}R$ is the Ricci scalar of the spacial part,
$K_{\text{s}}$ is the kinetic part of the spatial coordinates of the matter
content, and $V$ the matter potential.

There are many actions from which the WDW equation (1) derives as an equation
of motion. For example, let us consider a real scalar field $\Omega,$ leading
to the dynamics

$S_{3Q}=\int\text{d}\alpha\text{d}\phi\left[-\Psi^{\dagger}\square\Psi-\frac{1}{2}\Omega\square\Omega+\right.\\\
\left.+g_{3}\Omega\Psi^{\dagger}\Psi+V(\Omega^{2})\right],$ (8)

where $V(\Omega^{2})$ is the particular potential of the new field $\Omega$,
$g_{3}$ is a coupling constant and we get the relation $g_{3}\Omega=m^{2}$.
The field $\Omega$ is expected to be real since it is proportional to
$m^{2}(\alpha,\phi)$, which can be any real, possibly negative, number. The
new field thus has its own dynamics ruled by the equation of motion

$-\frac{1}{g_{3}}\left\\{\left[\partialderivative[2]{\alpha}-\partialderivative[2]{\phi}\right]\Omega+\partialderivative{V(\Omega^{2})}{\Omega}\right\\}=\Psi^{\dagger}\Psi.$
(9)

From this equation of motion it seems that there is a way to control the
unitarity of the wave function, since there is an explicit form of
$\absolutevalue{\Psi(\alpha,\phi)}^{2}$ given in terms of $\Omega$, but such
control is not perfect. For instance, let us consider the potential
$V(\Omega^{2})=M_{\Omega}\Omega^{2}/2$ and that the wave function of the
universe fulfills $\absolutevalue{\Psi(\alpha,\phi)}^{2}=C$, at all points of
the phase space $\\{\alpha,\phi\\}$, where $C$ is a positive constant. Eq. (9)
is then

$\left[\partialderivative[2]{\alpha}-\partialderivative[2]{\phi}\right]\Omega+M_{\Omega}\Omega=-g_{3}.$
(10)

Taking a look at Eq. (2), the limit at early times shows a vanishing function
for $m^{2}(\alpha,\phi)$ and its derivatives, hence $\Omega$ is also
vanishing. It is not in agreement with what we obtain from Eq. (10) since
$\Omega$ cannot be vanishing.

Figure 1: Interaction between universes without the mediation by the
$\Omega$-bosons (left) and with them (right) as derived from the action (8),
i. e. the second example. Time’s flow in this case implies a change in the
variable $\alpha,$ i. e. the scale factor.

Alternatively, it is natural to consider a complex scalar field $\Omega$. In
this case, the action reads

$S_{3Q}=\int\text{d}\alpha\text{d}\phi\Big{[}-\Psi^{\dagger}\square\Psi-\Omega^{\dagger}\square\Omega+\\\
+g_{4}\Omega^{\dagger}\Omega\Psi^{\dagger}\Psi+V(\absolutevalue{\Omega}^{2})\Big{]},$
(11)

from where we recognize $m^{2}=g_{4}\Omega^{\dagger}\Omega>0$. It is important
to notice that here the action is bounded from below, since
$m^{2}(\alpha,\phi)>0$, which was our weak condition to get a bounded action
(5), leading to a simplification in the physical interpretation. This scenario
constraints all universes which do not fulfill the condition. In theory, there
should not be any problem to describe the actual universe since the mass term
in Eq. (1) $m^{2}(\alpha,\phi)$ is always positive during standard
$\Lambda$CDM evolution [30]. The dynamics of the new field $\Omega^{\dagger}$
is now governed by the equation of motion

$\left[\partialderivative[2]{\alpha}-\partialderivative[2]{\phi}+g_{4}\Psi^{\dagger}\Psi\right]\Omega=-\partialderivative{V(\absolutevalue{\Omega}^{2})}{\Omega^{\dagger}},$
(12)

while $\Omega$ obeys the complex conjugated equation. Both equations for the
new particle are similar to a WDW equation (1), and coincide when
$V(\absolutevalue{\Omega}^{2})$ vanishes. Whether self-interactions of the
$\Omega$ field exist or not is impossible to say, and that is why we cannot
conclude that it is a universe-like field like $\Psi$. Even so, it
intentionally preserves the $U(1)$ symmetry found in the action (5) such that
the total electric charge and the number of particles continue to be
conserved. This example is clearly more natural than the first one. However,
in contrast with the latter model, it does not lead to a three-leg vertex.
Correspondingly, the extensively-studied picture of parent universes giving
birth to baby universes [31] is rendered impossible unless exotic ingredients
like axion-like particles are included [32].

Figure 2: Interaction between universes without the mediation by the
$\Omega$-bosons (left) and with them (right) as derived from the action (11),
i. e. the second example. Time’s flow in this case implies a change in the
variable $\alpha,$ i. e. the scale factor.

Nonetheless, most of the work done on Third Quantization is not spoiled since
the WDW equation (1) may be understood as an effective description of the
underlying field theory we propose. Note that due to the "coordinate"
dependence of the mass in the third quantized action (5), a nonperturbative
solution of the path integral was not achievable in the first place, seemingly
resulting in a hidden Fermi-like interaction provided on the left of Figures 1
and 2. These diagrams are completed by the introduction of the real scalar and
the complex scalar mediators to the right of Figures (1) and (2),
respectively. For instance, the entanglement entropy of the pair of universes
calculated in Refs. [16, 7, 17] remains unchanged for all models. However, the
spectra of the infinite universes created by dynamical variables in
minisuperspace [19, 20, 21, 22] square well with this new interpretation
because the background is flat, while an external field is held accountable
for this variation. The resulting field theory on flat spacetime, including a
source term, is thus equivalent to the canonical Third Quantization formalism
on a dynamical, i. e. expanding, minisuperspace. The only difference lies in
the interactions between universes and antiuniverses by virtue of a new field.
Hence, the concept of Third Quantization is carried through all the way
inasmuch as every field is quantized instead of treating some of them as
classical background.

##  III Conclusion

Recent work on interuniversal entanglement consistently points towards a time
evolution of corresponding entropy. This fact calls into question the concept
of noninteracting pairs of universes without background. We have shown how the
problem can be circumvented by introducing a new particle mediating
interactions between universes and possibly providing a heat bath for
decoherence. The resulting theory may be understood as a reformulation of
Third Quantization such that the difference to standard free scalar field
theory becomes manifest.

Although this formalism leads to the same predictions as standard Third
Quantization, it makes apparent that different universes do in fact interact,
which is the source for the variation of the interuniversal entanglement
entropy. An interaction between universes, in turn, makes these observable at
least in principle, thus possibly carrying the multiverse out of the realm of
the metaphysical. It would be interesting to quantify such observables in
future work.

##  IV Acknowledgements

S.B.B. would like to thanks L. M. Leza for the philosophical conversation that
planted the seed for the present work. Both authors express their gratitude to
S. Robles-Perez for his useful comments and corrections. This work has been
supported by the Polish National Research and Development Center (NCBR)
project UNIWERSYTET 2.0. STREFA KARIERY, POWR.03.05.00-00-Z064/17-00.

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