# Divergent geodesics in the Universal Teichmüller space

Xinlong Dong and Hrant Hakobyan  XD: Department of Mathematics, Kingsborough
CC - CUNY. Brooklyn, NY 11235<EMAIL_ADDRESS>HH: Department of
Mathematics, Kansas State University. Manhattan, KS 66502
<EMAIL_ADDRESS>
###### Abstract.

Thurston boundary of the universal Teichmüller space $T(\mathbb{D})$ is the
space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of
$\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of generalized Teichmüller
type if it shrinks the vertical foliation of a holomorphic quadratic
differential. We provide the first examples of generalized Teichmüller rays
which diverge near Thurston boundary $PLM_{bdd}(\mathbb{D})$. Moreover, for
every $k\geq 1$ we construct examples of rays with limit sets homeomorphic to
$k$-dimensional cubes. For the latter result we utilize the classical
Kronecker approximation theorem from number theory which states that if
$\theta_{1},\ldots,\theta_{k}$ are rationally independent reals then the
sequence $(\\{\theta_{1}n\\},\ldots,\\{\theta_{k}n\\})$ is dense in the
$k$-torus $\mathbb{T}^{k}$.

##  1\. Introduction

In this paper we continue the study of asymptotic behavor of geodesics in the
universal Teichmüller space $T(\mathbb{D})$ started in [HŠ16, HŠ18, HŠ21]. In
these works the main goal was to exhibit large families of geodesics which
converge at Thurston’s boundary $\partial_{\infty}T(\mathbb{D})$. The latter
can be identified with the space of projective classes of bounded geodesic
measured laminations of the hyperbolic plane $PML_{bdd}(\mathbb{D})$, cf.
[BŠ21]. In this paper we construct the first examples of divergent geodesics
in $T(\mathbb{D})$. Moreover, we show that the limit set of a divergent
geodesic in $PML_{bdd}(\mathbb{D})$ can be homeomorphic to a $k$-dimensional
cube $[0,1]^{k}$, for every $k\geq 1$.

For finite dimensional Teichmüller spaces the first examples of divergent
Teichmüller geodesics were given by Lenzhen in [Len08]. Limit sets of
Teichmüller and Weil-Petersson geodesics in the context of finite dimensional
Teichmüller spaces were actively investigated recently, see for instance
[BLMR1, BLMR2, BLMR3, CMW19, LLR18, LMR18]. Thus the present paper can be
thought of as a first step in obtaining analogous results for $T(\mathbb{D})$.
The techniques used in this paper are very different from the finite
dimensional ones and are more analytical in nature. Similar to [HŠ21] we rely
heavily on careful estimates of classical modulus for degenerating families of
curves in domains with chimneys. In [HŠ21] it was essentially shown that if
$D$ is a domain with chimneys s.t. the blowups of $D$ near boundary points
converge to a half plane, a complement of a quadrant, or a complement of a
half line then the geodesic in $T(\mathbb{D})$ corresponding to $D$ was
convergent. The idea behind constructing a divergent geodesic then is to
consider domains so that the blowups near a boundary point do not converge (in
Hausdorff topology). Specifically, we consider domains $D$ with infinitely
many chimneys accumulating to $\\{0\\}\times(0,\infty)$ so that on some scales
$D$ “looks like a half-plane” and on other scales it “looks like a quadrant”.
Interestingly, such a construction can yield both convergent as well as
divergent geodesics. Theorems 1.1 and 3.1 characterize which of the two
possibilities occurs depending on the widths and the relative positions of the
chimneys. The rest of this introduction describes our results in more detail.

###  1.1. Universal Teichmüller space and Thurston boundary

Let $\mathbb{D}$ and $\mathbb{S}^{1}$ denote the unit disk and the unit circle
in the complex plane $\mathbb{C}$, respectively. The _universal Teichmüller
space_ , denoted by $T(\mathbb{D})$, is defined as the collection of all
quasisymmetric (or qs) self-maps of $\mathbb{S}^{1}$, which fix $1,i$ and
$-1$. Universal Teichmüller space may be equipped with the Teichmüller metric
$d_{T}(\cdot,\cdot)$ as follows. Given $f,g\in T(\mathbb{D})$, the
_Teichmüller distanse_ between $f$ and $g$ is defined by
$d_{T}(f,g)=\frac{1}{2}\inf_{h}\log K_{h},$ where, the infimum is over all the
quasiconformal (or qc) extensions $h:\mathbb{D}\to\mathbb{D}$ of $g\circ
f^{-1}:\mathbb{S}^{1}\to\mathbb{S}^{1}$, and $K_{{h}}$ denotes the maximal
dilatation of the qc map ${h}$, see Section 2.1 for the definitions of these
terms.

Thurston boundary of $T(\mathbb{D})$, denoted by
$\partial_{\infty}T(\mathbb{D})$ was defined and studied in [Šar05, BŠ21]. In
particular, it was shown in [BŠ21] that $\partial_{\infty}T(\mathbb{D})$ can
be identified with the space of projective bounded measured laminations of the
unit disk $\mathbb{D}$, denoted by $PML_{bdd}(\mathbb{D})$.

In [HŠ16, HŠ18, HŠ21] Šarić and the second named author studied the asymptotic
behavior of geodesic rays in $T(\mathbb{D})$. Specifically, suppose $\varphi$
is a holomorphic quadratic differential $\varphi$ on $\mathbb{D}$. If for
every $t\in[0,1)$ the Beltrami differential
$\mu_{\varphi}(t)=t|\varphi|/\varphi$ is extremal (see Section 2.3 for the
definition of extremality) then the corresponding path

(1.1) $\displaystyle t\mapsto T_{\varphi}(t):=\left[\mu_{\varphi}(t)\right]$

in $T(\mathbb{D})$ is a geodesic ray, which will be called a _generalized
Teichmüller geodesic ray corresponding to $\varphi$._ If $\varphi$ is an
integrable holomorphic quadratic differential then $\mu_{\varphi}(t)$ is
uniquely extremal by [Str64] and hence $T_{\varphi}$ is a geodesic ray. We
call such a ray simply a _Teichmüller geodesic ray corresponding to
$\varphi$._

In [HŠ18] it was proved that _every_ Teichmüller geodesic ray in
$T(\mathbb{D})$ converges (in weak ∗ topology) to a unique point in
$\partial_{\infty}T(\mathbb{D})=PML_{bdd}(\mathbb{D})$, and that distinct
Teichmüller geodesic rays converge to distinct points in
$PML_{bdd}(\mathbb{D})$. In particular, there is an open and dense set of
geodesics in $T(\mathbb{D})$ which have distinct limits at Thurston boundary.
In [HŠ21] if was shown that for a large class of _generalized_ Teichmüller
geodesic rays the convergence to unique points of $PML_{bdd}(\mathbb{D})$
still holds, however, distinct geodesics can converge to the same point at
infinity. In view of the above it is natural to ask if there are divergent
geodesics in $T(\mathbb{D})$, and if so what are the possible sets in
$PML_{bdd}(\mathbb{D})$ which can occur as limit sets of such divergent
geodesics.

###  1.2. Main results

To describe our results more precisely we recall the construction of a natural
class of generalized Teichmüller rays. Suppose we are given a simply connected
domain $D\subsetneq\mathbb{C}$ with a chimney. This means that there is a non-
trivial finite interval $(a,b)\subset\mathbb{R}$ such that
$(a,b)\times(0,\infty)\subset D$ and
$(\\{a\\}\cup\\{b\\})\times(t,\infty)\subset\partial D$ for some $t>0$. Let
$\phi_{D}:\mathbb{D}\to D$ be a conformal map and consider the holomorphic
quadratic differential $\varphi_{D}=dw^{2}$, where $w(z)=\phi_{D}(z)$,
$z\in\mathbb{D}$. It turns out, see e.g. [HŠ21], that in this case Beltrami
differential $t|\varphi_{D}|/\varphi_{D}$ is extremal, and the path $t\mapsto
T_{\varphi_{D}}(t)=\left[t\frac{|\varphi_{D}|}{\varphi_{D}}\right]$ is a
generalized Teichmüller ray. Geometrically $t{|\varphi_{D}|}/{\varphi_{D}}$
can be thought of as the Beltrami coefficient corresponding to the vertical
compression map $(x,y)\mapsto\left(x,\frac{1-t}{1+t}y\right)$ of $D$, which
degenerates as $t\to 1$, see Section 2.4. Šarić and the second author
considered rays $\\{T_{\varphi_{D}}(t)\\}$ corresponding to domains with
finitely many chimneys or infinitely many chimneys
$(a_{i},b_{j})\times(0,\infty)$ so that the sequences $\\{a_{n}\\}$ and
$\\{b_{n}\\}$ do not have accumulation points in $\mathbb{R}$. They showed
that for such domains the corresponding generalized Teichmüller rays have
limits in Thurston’s boundary $\partial_{\infty}T(\mathbb{D})$, see [HŠ21,
Thm. 6.3].

In this paper we consider domains with accumulating chimneys. Specifically,
let $\\{x_{n}\\}_{n=0}^{\infty}$ be a strictly decreasing sequences of
positive numbers so that $x_{n}\to 0$ as $n\to\infty$. Letting $Q_{4}=\\{z:\Re
z>0,\Im z<0\\}$ denote the forth quadrant in $\mathbb{C}$, we define the
domain $W=W(\\{x_{n}\\})\subset\mathbb{C}$ as follows:

(1.2) $\displaystyle
W=Q_{4}\cup\bigcup_{n=0}^{\infty}((x_{2n+1},x_{2n})\times\mathbb{R}).$

In Theorem 3.1 we give a complete description of the asymptotic behavior of
the generalized Teichmüller ray corresponding to $W$, provided the sequence
$x_{n}$ converges to $0$ fast enough. In particular, we show that the ray
$t\mapsto T_{\varphi_{W}}(t)$ either converges or it diverges and its limit
set $\Lambda\subset PML_{bdd}(\mathbb{D})$ is homeomorphic to $[0,1]$. To
state our results more precisely we introduce some notation.

Given a geodesic $\gamma$ in the unit disk we denote by $\delta_{\gamma}$ the
Dirac mass at $\gamma$. For every chimney
$C_{n}=(x_{2n+1},x_{2n})\times\mathbb{R}$ we consider the point
$z_{n}\in\partial{\mathbb{D}}$, which corresponds to points in $W$ which are
“in $C_{n}$ and are near $\infty$”, i.e., if $z\to z_{n}$ then
$\varphi_{W}(z)\in C_{n}$ and $\Im(\varphi_{W}(z))\to\infty$. Let
$\gamma_{n}^{+}$ and $\gamma_{n}^{-}$ be the geodesics in $\mathbb{D}$
connecting the preimages of $x_{2n}$ and $x_{2n+1}$ to $z_{n}$, respectively.
Let $\lambda_{W}$ be the geodesic measured lamination supported on these
geodesics and giving each a unit mass, i.e.,
$\lambda_{W}=\sum_{n=0}^{\infty}(\delta_{\gamma_{n}^{+}}+\delta_{\gamma_{n}^{-}})$.
We also denote by $g$ the geodesic connecting $-1$ to $-i$. The following
result shows that the limit set of $t\mapsto T_{\varphi_{D}}(t)$ is either a
point or an interval.

######  Theorem 1.1.

Let $W$ be a domain defined as in (1.2) s.t.
$\frac{x_{n+1}}{x_{n}}\underset{n\to\infty}{\longrightarrow}0$. Then the limit
set $\Lambda\subset PLM_{bdd}(\mathbb{D})$ of the generalized Teichmüller ray
$t\mapsto T_{\varphi_{W}}(t)$ can be described as follows:

(1.3) $\displaystyle\Lambda=\\{\llbracket
s\delta_{g}+(2/3)\lambda_{W}\rrbracket:s\in[m,M]\\},$

where

(1.4)
$\displaystyle\begin{split}m&=1+\liminf_{n\to\infty}\sum_{i=0}^{2n+1}\frac{(-1)^{i}\log
x_{i}}{\log x_{2n+1}},\quad
M=1+\limsup_{n\to\infty}\sum_{i=0}^{2n+1}\frac{(-1)^{i}\log x_{i}}{\log
x_{2n}}.\end{split}$

In particular, $t\mapsto T_{\varphi_{W}}(t)$ diverges at
$\partial_{\infty}T(\mathbb{D})$ if and only if $m<M$.

Theorem 1.1 is an easy consequence of Theorem 3.1 and is proved after the
statement of the latter. Applying Theorem 1.1 to $x_{n}\asymp{1}/{n!}$ yields
an example of a convergent geodesic ray with the limit point
$\llbracket\frac{3}{2}\delta_{g}+\frac{2}{3}\lambda_{W}\rrbracket\in
PML_{bdd}(\mathbb{D})$, see Examples 3.3 and 3.4.

To obtain divergent geodesics we need to consider sequences $x_{n}$ which
converge to $0$ faster than $1/n!$. For instance, we have the following
corollary of Theorem 1.1, see Example 3.5.

######  Corollary 1.2.

Fix $p>1$. Let $x_{0}=1,x_{1}=a$ and $x_{n}=x_{n-1}^{p}$ for $n>1$
(equivalently $x_{n}=a^{p^{n-1}}$). Then $t\mapsto T_{\varphi_{W}}(t)$ is a
generalized Teichmüller geodesic ray which diverges at Thurston boundary with
a limit set given by (3.5), where $m=1+\frac{1}{p+1},$ and
$M=1+\frac{p}{p+1}$.

####  1.2.1. Moduli of famililes of curves

To prove Theorems 1.1 and 3.1 we restate them in terms of the limiting
behavior of moduli of families of curves.

Let $w=\phi(z)$ be the conformal map of the unit disk $\mathbb{D}$ onto $W$
such that $\phi^{-1}$ takes $w=0$ to $z=-1$, $w=1$ to $z=1$, and as $z$
approaches $-1$ we have that $\phi(z)$ tends to $\infty$ while staying in the
fourth quadrant of $\mathbb{C}$ , see Figure 1.1.

Given open disjoint arcs $I,J$ on the unit circle $\partial\mathbb{D}$, let
$\Gamma_{I,J}=\Gamma(I,J;\mathbb{D})$ be the curve family connecting $I$ to
$J$ in $\mathbb{D}$. For every $\varepsilon>0$ we denote by $V_{\varepsilon}$
the vertical compession map $(x,y)\mapsto(x,\varepsilon y)$. Observe that if
$\varepsilon=\varepsilon(t)=\frac{1-t}{1+t}$, then $\varepsilon\to 0$ as $t\to
1$ , that is, as the corresponding point $[t|\phi_{W}|/\phi_{W}]\in
T(\mathbb{D})$ leaves every compact subset of $T(\mathbb{D})$.

Let $\Gamma^{\varepsilon}_{I,J}=V_{\varepsilon}(\phi(\Gamma_{I,J}))$. Using
the connection between the Liouville measure and moduli of curve families, see
Lemma 2.5 or [HŠ16, Lemma 3.3], understanding the behavior of the geodesic
$t\mapsto T_{\phi_{W}}(t)$ is equivalent to understanding the behavior of
$\mathrm{mod}\Gamma^{\varepsilon}_{I,J}$ for different choices of pairs of
arcs $I,J\in\mathbb{S}^{1}$, as $\varepsilon\to 0$. Namely, to show that a
point $\llbracket\lambda\rrbracket\in PML_{bdd}(\mathbb{D})$ is a limit point
of the generalized Teichmüller geodesic $t\mapsto T_{\phi_{W}}(t)$ it is
sufficient to show that there is a sequence $\varepsilon_{n}\to 0$ so that an
appropriate rescaling of the sequence
$\mathrm{mod}\Gamma^{\varepsilon_{n}}_{I,J}$ converges to $\lambda(I\times
J)$.

It follows essentially from [HŠ21] that if the intersection of the box
$I\times J$ and geodesic lamination
$\mathfrak{L}=\\{g\\}\cup\bigcup_{n=0}^{\infty}\\{\gamma_{n}^{+},\gamma_{n}^{-}\\}$
is empty then

$\lim_{\varepsilon\to
0}\frac{\mathrm{mod}\Gamma^{\varepsilon}_{I,J}}{\frac{1}{\pi}\log\frac{1}{\varepsilon}}=0.$

On the other hand, if $(I\times J)\cap\mathfrak{L}$ contains a single geodesic
from $\mathfrak{L}$ that is not $g$ then

$\lim_{\varepsilon\to
0}\frac{\mathrm{mod}\Gamma^{\varepsilon}_{I,J}}{\log\frac{1}{\varepsilon}}=\frac{2}{3}.$

If $(I\times J)\cap\mathfrak{L}=g$ we prove the following.

######  Lemma 1.3.

Suppose $W$ satisfies the conditions of Theorems 1.1 (or Theorem 3.1). If
$(I\times J)\cap\mathfrak{L}=g$ then

(1.5) $\displaystyle\liminf_{\varepsilon\to
0}\frac{\mathrm{mod}\Gamma^{\varepsilon}_{I,J}}{\frac{1}{\pi}\log\frac{1}{\varepsilon}}=m,\quad\limsup_{\varepsilon\to
0}\frac{\mathrm{mod}\Gamma^{\varepsilon}_{I,J}}{\frac{1}{\pi}\log\frac{1}{\varepsilon}}=M.$

Lemma 1.3 is a key result used in the proof of Theorems 1.1 and 3.1. It
follows from Proposition 4.2 and Lemma 5.1.

####  1.2.2. Divergent geodesics with higher dimensional limit sets

Using Theorem 1.1 we construct for every $k\in\mathbb{N}$ a domain
$\mathcal{W}_{k}$ such that the limit set corresponding to the ray $t\to
T_{\varphi_{\mathcal{W}_{k}}}(t)$ is homeomorphic to $[0,1]^{k}$. For
instance, for $k=2$ we have the following result, which follows from Theorem
6.1.

Figure 1.1. Supports of the limiting geodesic measured laminations
corresponding to the divergent geodesic rays constructed in Theorems 1.1 and
1.4.

######  Theorem 1.4.

Fix $a\in(0,1)$, and $p,q>1$ so that $\log p/\log q$ is irrational. Let
$x_{n}=a^{p^{n-1}}$, $x_{n}^{\prime}=3-a^{q^{n-1}}$, and
$C_{n},C_{n}^{\prime}$ be chimneys over the intervals $(x_{2n+1},x_{2n})$ and
$(x^{\prime}_{2n},x_{2n+1}^{\prime})$, respectively. Then the generalized
Teichmüller ray $t\mapsto T_{\varphi_{\mathcal{W}_{p,q}}}(t)$ corresponding to
the domain

$W_{p,q}=\\{z\,:\,0<\Re(z)<3,\,\Im(z)>0\\}\cup\bigcup_{n=1}^{\infty}C_{n}\cup
C^{\prime}_{n}$

diverges at $\partial_{\infty}T(\mathbb{D})$ and its limit set $\Lambda\subset
PML_{bdd}(\mathbb{D})$ in Thurston boundary is homeomorphic to $[0,1]^{2}$.

A key result used in the proof of Theorem 6.1 is the dynamical fact that every
orbit $\\{T_{\theta}^{\circ n}(x)\\}_{n=1}^{\infty}$ of the irrational
rotation $T_{\theta}:x\mapsto\\{\theta+x\\}$ is dense in the circle
$\mathbb{R}/\mathbb{Z}$, where $\theta=\log p/\log q$ and $\\{y\\}=y-\lfloor
y\rfloor$ denotes the fractional part of a real number $y$.

To prove a higher dimensional version of Theorem 1.4, see Theorem 6.3, we
construct domains with $k$ families of chimneys accumulating to distinct half-
lines corresponding to sequences of the form
$\\{a^{p_{i}^{n}}\\}_{n=1}^{\infty},i\in\\{1,\ldots,k\\},$ such that the
numbers $\log p_{1},\ldots,\log p_{k}$ are rationally independent. We use a
well known approximation theorem of Kronecker, which generalizes the above
mentioned fact about the density of the orbits of an irrational rotation of
$\mathbb{S}^{1}$ to higher dimensional tori
$\mathbb{T}^{k}=\mathbb{S}^{1}\times\ldots\times\mathbb{S}^{1}$.

In Theorems 1.1 and 1.4 if
$\llbracket\lambda_{1}\rrbracket,\llbracket\lambda_{2}\rrbracket\in\Lambda$
then the supports of $\lambda_{1}$ and $\lambda_{2}$ are the same. This raises
the following question.

######  Question 1.5.

Is there a limit set $\Lambda\subset PML_{bdd}(\mathbb{D})$ and geodesic
measured laminations $\lambda_{1}$ and $\lambda_{2}$, so that
$\llbracket\lambda_{1}\rrbracket,\llbracket\lambda_{2}\rrbracket\in\Lambda$
and the supports of $\lambda_{1}$ and $\lambda_{2}$ are distinct.

It would be interesting to undestand the possible topology of limit sets of
geodesic rays in $T(\mathbb{D})$. For instance, even the following basic
question seems to be open.

######  Question 1.6.

Is there a Teichmüller geodesic in $T(\mathbb{D})$ so that the limit set
$\Lambda\subset PML_{bdd}(\mathbb{D})$ has nontrivial topology (e.g.
$\pi_{1}(\Lambda)\neq\\{0\\}$)?

The rest of this paper is organized as follow. In Section 2 we provide the
necessary definitions, notation, and some auxiliary results. In Section 3 we
state Theorem 3.1, prove Theorem 1.1 and provide several explicit examples of
divergent and convergent geodesics in $T(\mathbb{D})$. Sections 4 and 5 are
the technical core of the paper, which are devoted to the proof of Theorem
3.1. In Section 6 we prove the existence of generalized Teichmüller geodesics
with higher dimensional limit sets.

##  2\. Background

###  2.1. Visual boundary of the universal Teichmüller space

If $E$ is a subset of $\mathbb{S}^{1}$ we will denote by $|E|$ the arclength
(or Hausdorff $1$-measure) of $E$.

A homeomorphism $h:\mathbb{S}^{1}\to\mathbb{S}^{1}$ is said to be
_quasisymmetric_ if there is a constant $1\leq M<\infty$ such that

(2.1) $\displaystyle\frac{1}{M}\leq\frac{|h(I)|}{|h(J)|}\leq M,$

for all circular arcs $I,J$ in $\mathbb{S}^{1}$ with a common boundary point
and disjoint interiors such that $|I|=|J|$. A homeomorphism is quasisymmetric
if and only if it extends to a quasiconformal map of the unit disk.

The _universal Teichmüller space_ , denoted by $T(\mathbb{D})$, consists of
all quasisymmetric $h:\mathbb{S}^{1}\to\mathbb{S}^{1}$, which fix $1,i$ and
$-1$. The _Teichmüller distanse_ between elements $f$ and $g$ in
$T(\mathbb{D})$ is defined as follows.:

$\displaystyle d_{T}(f,g)=\frac{1}{2}\inf\log K_{g\circ f^{-1}},$

where, given a quasisymmetric mapping $h$ of $\mathbb{S}^{1}$, we denote

$\displaystyle
K_{h}=\inf\\{K_{\tilde{h}}\,|\,\tilde{h}:\mathbb{D}\to\mathbb{D}\mbox{ is a
quasiconformal extension of }h\\},$

with $K_{\tilde{h}}$ denoting the maximal dilatation of the quasiconformal map
$\tilde{h}$ extending $h$ to $\mathbb{{D}}$. We refer to [Ahl06] for
background on quasisymmetric and quasiconformal maps and, in particular, for
the definitions of the maximal dilatation of a quasiconformal map and
quasiconformal extensions.

The universal Teichmüller space $T(\mathbb{D})$ may also be defined as the set
of equivalence classes of Beltrami coefficients $[\mu]\in B_{1}/\sim$, where
$B_{1}$ is the unit ball in $L_{\infty}(\mathbb{D})$, and $\mu\sim\nu$
whenever the corresponding quasiconformal mappings
$f^{\mu},f^{\nu}:\mathbb{D}\to\mathbb{D}$ coincide on the boundary circle,
i.e. $f^{\mu}|_{\mathbb{S}^{1}}=f^{\nu}|_{\mathbb{S}^{1}}$, see [GL00]. Given
two Beltrami coefficients $\mu_{0}$ and $\nu_{0}$ the Teichmüller distance
between $[\mu_{0}]$ and $[\nu_{0}]$ in $T(\mathbb{D})$ is defined as follows

(2.2) $\displaystyle
d_{T}([\mu_{0}],[\nu_{0}])=\frac{1}{2}\inf_{\begin{subarray}{c}\mu\in[\mu_{0}]\\\
\nu\in[\nu_{0}]\end{subarray}}\log\frac{1+\left\|\frac{\mu-\nu}{1-\bar{\mu}\nu}\right\|_{\infty}}{1-\left\|\frac{\mu-\nu}{1-\bar{\mu}\nu}\right\|_{\infty}}.$

Given a quasisymmetric mapping $h:\mathbb{S}^{1}\to\mathbb{S}^{1}$, a
quasiconformal mapping $f:\mathbb{D}\to\mathbb{D}$ continuously extending $h$
is called an extremal quasiconformal mapping (for its boundary values) if it
has the smallest maximal dilatation $K_{f}$ among all such extensions of $h$
to $\mathbb{D}$.

A Beltrami coefficient $\mu$ is said to be extremal or uniquely extremal if
$\|\mu\|_{\infty}\leq\|\nu\|_{\infty}\mbox{ or
}\|\mu\|_{\infty}<\|\nu\|_{\infty},$ respectively, whenever $\nu\sim\mu$ and
$\nu\neq\mu$. If $\mu$ is an extremal Beltrami coefficient then

$\displaystyle\mu_{s}=\frac{(1+|\mu|)^{s}-(1-|\mu|)^{s}}{(1+|\mu|)^{s}+(1-|\mu|)^{s}}\frac{\mu}{|\mu|},$

is also extremal for every $s\in[0,\infty)$, see [FM07]. Moreover, with the
notation as above we have

$\displaystyle\mathrm{dist}_{T}([0],[\mu_{s}])=s\cdot\mathrm{dist}_{T}([0],[\mu]).$

Therefore, the path $s\mapsto[\mu_{s}]$ is a geodesic ray in the Teichmüller
metric in $T(\mathbb{D})$, i.e. it is a geodesic path starting at $[0]$,
passing through $[\mu]$ and leaving every compact set in $T(\mathbb{D})$ as
$s\to\infty$. The collection of all geodesic rays in $T(\mathbb{D})$ is the
_visual boundary_ of the universal Teichmüller space, see [HŠ21].

###  2.2. Thurston boundary of $T(\mathbb{D})$

In [Bon88] Thurston’s boundary of the Teichmüller space $T(S)$ of a closed
surface $S$ was defined by embedding $T(S)$ into the space of geodesic
currents on $S$. In [Šar05, Šar15, BŠ21] the definition for a Thurston type
boundary of Techmüller spaces was given for all Riemann surfaces, and in
particular for $T(\mathbb{D})$. We briefly recall some of the relevant
notation, see [HŠ21].

The space $G(\mathbb{D})$ of oriented geodesics on $\mathbb{D}$ can be
identified with $\mathbb{S}^{1}\times\mathbb{S}^{1}\setminus diag$. Observe
that a quasisymmetric mapping $f:\mathbb{S}^{1}\to\mathbb{S}^{1}$ induces a
self-map of the space of geodesics $G(\mathbb{D})$ by mapping a hyperbolic
geodesic $\gamma_{(x,y)}$ to $\gamma_{(f(x),f(y))}$, for every pair of
distinct point $x,y\in\mathbb{S}^{1}$. We denote by $\tilde{f}$ the mapping
induced by $f$ on the space of geodesics.

A geodesic current is a Radon measure on $G(\mathbb{D})$. The Liouville
current $\mathcal{L}$ is a geodesic current such that for every Borel set
$A\subset\mathbb{S}^{1}\times\mathbb{S}^{1}\setminus diag$ we have

$\mathcal{L}(A)=\int_{A}\frac{|dx||dy|}{|x-y|^{2}}.$

An easy calculation shows that for a box of geodesics $A=[a,b]\times[c,d]$ we
have

$\mathcal{L}([a,b]\times[c,d])=\log\frac{(a-c)(b-d)}{(a-d)(b-c)}.$

The universal Teichmüller space $T(\mathbb{D})$ can be embedded into the space
$\mathcal{C}(\mathbb{D})$ of geodesic currents via the _Liouville embedding_
by setting

$\mathscr{L}(f)=(\tilde{f})^{*}(\mathcal{L})\in\mathcal{C}(\mathbb{D}),$

where the right hand side denotes the pullback of the Liouville measure by
$\tilde{f}$. Moreover, whenever $f$ is a quasisymmetric then $\mathscr{L}(f)$
is a _bounded geodesic current_ in the sense that
$\sup_{A}\mathscr{L}(f)(A)<\infty$, where the supremum is over all geodesic
boxes $A$ such that $\mathcal{L}(A)=\log 2$ (thus $A$ is a Mobius image of the
box $[1,i]\times[-1,-i]$).

Liouville embedding $\mathscr{L}$ is in fact a homeomorphism of
$T(\mathbb{D})$ onto its image in $\mathcal{C}(\mathbb{D})$ equipped with the
uniform weak* topology, see [MŠ12, Šar15]. As a set _Thurston boundary of
$T(\mathbb{D})$_, which is denoted by $\partial_{\infty}T(\mathbb{D})$, is
defined as the collection of asymptotic rays to $\mathscr{L}(T(\mathbb{D}))$
in $\mathcal{C}(\mathbb{D})$. Equivalently, $\partial_{\infty}T(\mathbb{D})$
can be identified with the space of projective classes of bounded measured
laminations on $\mathbb{D}$, or $PML_{bdd}(\mathbb{D})$, see [Šar05, Šar15].

###  2.3. Generalized Teichmüller rays in $T(\mathbb{D})$.

In this paper we continue the study of behavior of geodesic rays in
$T(\mathbb{D})$ as they approach infinity, i.e. leave every compact subset of
$T(\mathbb{D})$ started in [HŠ16, HŠ18, HŠ21]. Specifically, let $\varphi$ be
a holomorphic quadratic differential on $\mathbb{D}$. Suppose that
$\mu_{\varphi}(t):=t|\varphi|/\varphi$ is an extremal Beltrami differential
for some (or equivalently all) $t\in(0,1)$. By (2.2) we have

$\displaystyle
d_{T}([0],[\mu_{\varphi}(t)])=\frac{1}{2}\log\frac{1+\|\mu_{\varphi}(t)\|_{\infty}}{1-\|\mu_{\varphi}(t)\|_{\infty}}=\frac{1}{2}\log\frac{1+t}{1-t}.$

Therefore, if $0<s<t<1$ then by (2.2) again we have

$d_{T}([\mu_{\varphi}(s)],[\mu_{\varphi}(t)])=d_{T}([0],[\mu_{\varphi}(t)])-d_{T}([0],[\mu_{\varphi}(s)]).$

Hence the path

(2.3) $\displaystyle t\mapsto[t{|\varphi|}/{\varphi}]$

is a geodesic ray in the Teichmüller metric in $T(\mathbb{D})$. We say that
the path (2.3) is a _generalized Teichmüller ray_ if the Beltrami coefficient
$t\frac{|\varphi|}{\varphi}$ is extremal for some (or equivalently all)
$t\in(0,1)$.

###  2.4. Generalized Teichmüller rays and vertical compression

A natural class of generalized Teichmüller rays can be obtained as follows.
Given a simply connected domain $D\subsetneq\mathbb{C}$ and a conformal map
$\phi_{D}:\mathbb{D}\to D$ consider the holomorphic quadratic differential
$\varphi_{D}=dw^{2}$, where $w(z)=\phi_{D}(z)$, $z\in\mathbb{D}$. If the
Beltrami differential $t|\varphi_{D}|/\varphi_{D}$ happens to be extremal for
some $t\in(0,1)$ then the generalized Teichmüller ray
$T_{\varphi_{D}}(t)=[\mu_{\varphi_{D}}(t)]$ for $\varphi=\varphi_{D}$ can
alternatively be described as follows, see [HŠ21]

For every $0<\varepsilon\leq 1$, let $V_{\varepsilon}$ be the vertical
compression map of the plane, i.e.

$\displaystyle V_{\varepsilon}(x,y)=(x,\varepsilon y),$

and let $\phi_{D,\varepsilon}$ be a conformal map of the disc $\mathbb{D}$
onto $V_{\varepsilon}(D)$ and let $\phi_{D}:=\phi_{D,1}:\mathbb{D}\to D.$ The
mapping

$\psi_{D,\varepsilon}:=\phi_{D,\varepsilon}^{-1}\circ
V_{\varepsilon}\circ\phi_{D}$

is a quasiconformal mapping of $\mathbb{D}$ onto itself with the Beltrami
differential given by
$\left(\frac{1-\varepsilon}{1+\varepsilon}\right)\frac{|\varphi_{D}|}{\varphi_{D}}$
and the maximal dilatation $1/\varepsilon$. If $V_{\varepsilon}$
(respectively, $\phi_{D,\varepsilon}^{-1}\circ V_{\varepsilon}\circ\phi_{D}$)
is an extremal quasiconformal map for its boundary values on $\partial{D}$
(respectively, on $\mathbb{S}^{1}$), then the path

$\varepsilon\mapsto\tau_{\varepsilon,{D}}=\left[\left(\frac{1-\varepsilon}{1+\varepsilon}\right)\frac{|\varphi_{D}|}{\varphi_{D}}\right]$

with $\varepsilon$ decreasing from $1$ to $0$ is a generalized Teichmüller ray
in $T(\mathbb{D})$, and we have

$\displaystyle\mathrm{dist}([0],\tau_{\varepsilon,D})=\frac{1}{2}\log\varepsilon^{-1}\underset{\varepsilon\to
0}{\longrightarrow}\infty.$

###  2.5. From Teichmüller rays to conformal modulus

Suppose $D$ is a domain in the complex plane as above, and
$\varepsilon\mapsto\tau_{\varepsilon,D}$ is the corresponding Teichmüller ray.
Abusing the notation slightly, we will denote by $\psi_{D,\varepsilon}$ (or
$\psi_{\varepsilon}$), if $D$ is clear from the context) the corresponding
quasisymmetric mapping of $\mathbb{S}^{1}$. Therefore, the Teichmüller ray
corresponding to $D$ (or $\varphi_{D}$) can also be represented as follows:

$\displaystyle\varepsilon\mapsto\psi_{\varepsilon}=\psi_{D,\varepsilon}.$

To study the asymptotic behavior of the geodesic
$\varepsilon\mapsto\psi_{\varepsilon}$ in $T(\mathbb{D})$ near
$\partial_{\infty}T(\mathbb{D})$, i.e., as $\varepsilon\to 0$, in [HŠ16, HŠ18,
HŠ21] the authors used Liouville embedding of the Teichmüller space and a
connection between Liouville measure and the classical modulus of families and
curves.

####  2.5.1. Limit sets of geodesic rays in $T(\mathbb{D})$

Let $\lambda$ be a bounded measured lamination of the Hyperbolic plane
$\mathbb{D}$. We will denote by $\llbracket\lambda\rrbracket$ the projective
class of $\lambda$. We say that a bounded projective measured lamination
$\llbracket{\lambda}\rrbracket\in PML_{bdd}(\mathbb{D})$ is a _limit point_ of
a geodesic ray $\varepsilon\mapsto\psi_{D,\varepsilon}$ if there is a sequence
of positive numbers $\\{\varepsilon_{i}\\}_{i=1}^{\infty}$ approaching $0$
such that

(2.4)
$\displaystyle\llbracket{\mathscr{L}(\psi_{D,\varepsilon_{i}})}\rrbracket\underset{i\to\infty}{\longrightarrow}\llbracket{\lambda}\rrbracket,$

in the weak* topology. The _limis set_ of the generalized Teichmüller ray
$\varepsilon\mapsto\psi_{D,\varepsilon}$ is the collection of all its limit
points in $PML_{bdd}(\mathbb{D})$.

To obtain a more explicit expression of the limit points of the rays in
$T(\mathbb{D})$ we recall that for a given box $[a,b]\times[c,d]\in
G(\mathbb{D})$ and $\varepsilon>0$ we have

$\displaystyle\mathscr{L}(\psi_{\varepsilon})([a,b]\times[c,d])$
$\displaystyle=\tilde{(\psi_{\varepsilon})}^{*}(\mathcal{L})([a,b]\times[c,d])$
$\displaystyle=\mathcal{L}(\psi_{\varepsilon}([a,b])\times\psi_{\varepsilon}([c,d])).$

Denoting $x^{\varepsilon}:=\psi_{\varepsilon}(x)$ for every
$x\in\mathbb{S}^{1}$ we can write the equality above as

$\displaystyle\mathscr{L}(\psi_{\varepsilon})([a,b]\times[c,d])=\mathcal{L}([a^{\varepsilon},b^{\varepsilon}]\times[c^{\varepsilon},d^{\varepsilon}]).$

Thus, to find the limit points of a generalized Teichmüller geodesic we need
to find the asymptotic behavior of
$\mathcal{L}([a^{\varepsilon},b^{\varepsilon}]\times[c^{\varepsilon},d^{\varepsilon}])$
as $\varepsilon$ approaches $0$. To do this we recall the notion of modulus of
families of curves.

####  2.5.2. From Liouville measure to conformal modulus.

Let $\Gamma$ be a family of curves in a domain $\Omega\subset\mathbb{C}$. A
non-negative Borel function $\rho$ on $\Omega$ is called _admissible for
$\Gamma$_, if $l_{\rho}(\gamma):=\int_{\gamma}\rho(z)|dz|\geq 1$ for every
$\gamma\in\Gamma$. Conformal modulus of $\Gamma$ then is defined by

$\displaystyle\mathrm{mod}\Gamma=\inf_{\rho\mbox{\tiny{ is
$\Gamma$-adm}}}\iint_{\Omega}\rho^{2}dxdy,$

where the infimum is over all $\Gamma$ admissible metrics $\rho$.

Lemmas 2.1, 2.2 and 2.3 below, summarize some of the main properties of the
modulus, which we will use below, see [GM05, LV73, Väi71].

If $\Gamma_{1}$ and $\Gamma_{2}$ are curve families in $\mathbb{C}$, we will
say that $\Gamma_{1}$ _overflows_ $\Gamma_{2}$ and will write
$\Gamma_{1}>\Gamma_{2}$ if every curve $\gamma_{1}\in\Gamma_{1}$ contains some
curve $\gamma_{2}\in\Gamma_{2}$.

######  Lemma 2.1.

Let $\Gamma_{1},\Gamma_{2},\ldots$ be curve families in $\mathbb{C}$. Then

  * 1.

Monotonicity: If $\Gamma_{1}\subset\Gamma_{2}$ then
$\mathrm{mod}(\Gamma_{1})\leq\mathrm{mod}(\Gamma_{2})$.

  * 2.

Subadditivity:
$\mathrm{mod}(\bigcup_{i=1}^{\infty}\Gamma_{i})\leq\sum_{i=1}^{\infty}\mathrm{mod}(\Gamma_{i}).$

  * 3.

Overflowing: If $\Gamma_{1}>\Gamma_{2}$ then
$\mathrm{mod}\Gamma_{1}\leq\mathrm{mod}\Gamma_{2}$.

  * 4.

Conformal invariance: If $f:\Omega\to\Omega^{\prime}$ is a conformal map then

$\mathrm{mod}f(\Gamma)=\mathrm{mod}\Gamma$

for any curve family $\Gamma$ in $\Omega$, where $f(\Gamma)$ denotes the
family of curves $\\{f(\gamma):\gamma\in\Gamma\\}$ in $\Omega^{\prime}$.

The next two examples of curve families are fundamental and will be used
repeatedly throughout the paper, see [Ahl06].

######  Lemma 2.2 (Modulus of a rectangle).

Let $R=[0,l]\times[0,w]$ and $\Gamma$ be the family of curves in $R$
connecting the vertical sides, i.e., $\\{0\\}\times[0,w]$ and
$\\{l\\}\times[0,w]$. Then $\mathrm{mod}\Gamma=w/l$.

######  Lemma 2.3 (Modulus of an annulus).

Suppose $A=\\{z:0<r_{1}<|z|<r_{2}\\}$. Let $\Gamma_{A}$ and
$\Gamma_{A}^{\prime}$ be the families of curves in $A$ separating and
connecting the two boundary components of $A$, respectively. Then

$\displaystyle\mathrm{mod}\Gamma_{A}=\frac{\log\frac{r_{2}}{r_{1}}}{2\pi},\qquad\mathrm{mod}\Gamma_{A}^{\prime}=\frac{2\pi}{\log\frac{r_{2}}{r_{1}}}.$

The following estimate for modulus will be used repeatedly below.

Given two continua $E$ and $F$ in $\mathbb{C}$ we denote

$\displaystyle\Delta(E,F):=\frac{\mathrm{dist}(E,F)}{\min\\{\mathrm{diam}E,\mathrm{diam}F\\}},$

and call $\Delta(E,F)$ the relative distance between $E$ and $F$ in
$\mathbb{C}$.

######  Lemma 2.4 (cf. [HŠ18]).

For every pair of continua $E,F\subset\mathbb{C}$ we have

(2.5)
$\displaystyle\mathrm{mod}(E,F;\mathbb{C})\leq\pi\left(1+\frac{1}{2\Delta(E,F)}\right)^{2}.$

In particular if $E_{n}$ and $F_{n}$ are such that $\Delta(E_{n},F_{n})$ is
bounded away from $0$ then $\mathrm{mod}(E_{n},F_{n};\mathbb{C})$ is bounded
above.

The following result, connects Liouville measure and the moduli of curve
fmilies and is central for our analysis.

######  Lemma 2.5 (See Lemma 3.3 in [HŠ16]).

Let $(a,b,c,d)$ be a quadruple of points on $\mathbb{S}^{1}$ in the
counterclockwise order, and let $\Gamma_{[a,b]\times[c,d]}$ be the family of
curves in $\mathbb{D}$ connecting $[a,b]$ to $[c,d]$. Then

(2.6)
$\displaystyle\mathrm{mod}(\Gamma_{[a,b]\times[c,d]})-\frac{1}{\pi}\mathcal{L}([a,b]\times[c,d])-\frac{2}{\pi}\log
4\to 0$

as $\mathrm{mod}(\Gamma_{[a,b]\times[c,d]})\to\infty$, where $\mathcal{L}$ is
the Liouville measure.

######  Remark 2.6.

Note that by equation (2.6) we have that
$\mathrm{mod}(\Gamma_{[a,b]\times[c,d]})\to\infty$ if and only if
$\mathcal{L}([a,b]\times[c,d])\to\infty$. Therefore it is enough to consider
the asymptotic behavior of the modulus in order to find the asymptotic
behavior of the Liouville measure.

We next give a criterion from [HŠ21] which will help identify which elements
of $PLM_{bdd}(\mathbb{D})$ can occur as limit points at infinity of
generalized Teichmüller rays in $T(\mathbb{D})$.

######  Definition 2.7.

A set $\mathcal{B}$ of boxes of geodesics is said to be dense among all boxes
of geodesics if for any box $[a,b]\times[c,d]\subset G(\mathbb{D})$ there
exists a sequence $\\{[a_{n},b_{n}]\times[c_{n},d_{n}]\\}_{n}$ in
$\mathcal{B}$ such that $a_{n}\to a$, $b_{n}\to b$, $c_{n}\to c$ and $d_{n}\to
d$ as $n\to\infty$.

######  Corollary 2.8.

Let $D\subset\mathbb{C}$ be a domain such that
$\varepsilon\mapsto\psi_{D,\varepsilon}$ is a generalized Teichmüller ray.
Suppose that there is a bounded geodesic current $\lambda$, a function
$m=m(\varepsilon)\to\infty,$ as $\varepsilon\to 0,$ and a sequence
$\varepsilon_{i}\to 0$ such that the limit

(2.7)
$\displaystyle\lambda([a,b]\times[c,d])=\lim_{i\to\infty}\frac{\mathrm{mod}V_{\varepsilon_{i}}(\phi_{D}(\Gamma_{[a,b]\times[c,d]}))}{m(\varepsilon_{i})}$

exists for a box of geodesics $[a,b]\times[c,d]$. Then
$\llbracket\lambda\rrbracket$ is a limit point of the generalized Teichmüller
ray $\varepsilon\mapsto\psi_{D,\varepsilon}$ in the weak* topology if and only
if (2.7) holds for a dense set $\mathcal{B}$ of boxes of geodesics in
$G(\mathbb{D})$.

We note that in [HŠ21] the sufficiency of the condition (2.7) for a dense set
of boxes was proved for the entire ray
$\varepsilon\mapsto\psi_{D,\varepsilon}$ rather than for a sequence of points
$\psi_{D,\varepsilon_{i}}$, but the same proof gives the more general result
above. On the other hand, the necessity of (2.7) follows from (2.4) by using
(2.6) and recalling the relevant definitions.

In some cases it is easier to find the limiting geodesic measured lamination
using a localization result which we formulate next.

For $\eta\in(0,\pi)$ we will denote by $B_{\eta}(\gamma)$ the $\eta$-box
containing $\gamma$, which may be thought of as the $\eta$-neighborhood of
$\gamma$. More precisely, if $\gamma=\gamma_{(a,b)}\in G(D)$ for some
$a=e^{i\alpha}$ and $b=e^{i\beta}$ in $\mathbb{S}^{1}$, we let

$\displaystyle
B_{\eta}(\gamma)=(e^{i(\alpha-\eta)},e^{i(\alpha+\eta)})\times(e^{i(\beta-\eta)},e^{i(\beta+\eta)}).$

With this notation we have the following consequence of Corollary 2.8, which
is an analogue of Lemma 6.6 in [HŠ21] and can be proved by selecting
appropriately the family $\mathcal{B}$ of boxes and using the basic properties
of the modulus from Lemma 2.1.

######  Lemma 2.9.

Let $D\subset\mathbb{C}$ be a domain such that
$\varepsilon\mapsto\psi_{D,\varepsilon}$ is a generalized Teichmüller
geodesic, and $\\{\varepsilon_{i}\\}_{i=0}^{\infty}$ is a positive sequence
which converges to $0$ as $i\to\infty$. Suppose for every $\gamma\in
G(\mathbb{D})$ there is a positive number $\eta=\eta_{\gamma}>0$ s.t. the
limit

(2.8) $\displaystyle
m(\gamma)=m_{\eta}(\gamma):=\lim_{i\to\infty}\frac{\mathrm{mod}V_{\varepsilon_{i}}(\phi(\Gamma_{B_{\eta}(\gamma)}))}{\log\frac{1}{\varepsilon_{i}}}$

exists for all $0<\eta<\eta_{\gamma}$ and is independent of $\eta$. If there
is a countable collection of geodesics $\\{\gamma_{j}\\}_{j=1}^{\infty}$ s.t.
$m(\gamma)>0$ if and only if $\gamma=\gamma_{j}$ for some $j\geq 1$, then

$\displaystyle\tau_{\varepsilon_{i},D}\underset{i\to\infty}{\longrightarrow}\left\llbracket\sum_{j=1}^{\infty}m(\gamma_{j})\delta_{\gamma_{j}}\right\rrbracket.$

###  2.6. Chimney domains and generalized Teichmüller rays

We say that $\Omega\subset\mathbb{C}$ is a _domain with a chimney_ if $\Omega$
contains a product subset $C_{(a,b)}:=(a,b)\times(0,\infty)$ such that
$\partial C_{(a,b)}\cap\\{z:\Im z>\alpha\\}\subset\partial\Omega$ for some
$\alpha\geq 0$.

If $\Omega$ is a domain with a chimney then by Theorem 6.1 in [HŠ21] the
vertical compression map $V_{\varepsilon}:(x,y)\mapsto(x,\varepsilon y)$ is an
extremal quasiconformal map for its boundary values on $\partial{\Omega}$.
Therefore, for any domain $\Omega$ with a chimney the path
$\varepsilon\mapsto\psi_{\Omega,\varepsilon}$ defined in Section 2.4 is a
generalized Teichmüller geodesic. Hence, to conclude that $\lambda\in
PML_{bdd}(\mathbb{D})$ is a limit point of the generalized Teichmüller ray
$\varepsilon\mapsto\psi_{\Omega,\varepsilon}$ for a chimney domain $\Omega$,
by Corollary 2.8 it is enough to verify that equality (2.7) holds for a dense
set of boxes of geodesics.

##  3\. Main result and some examples

Let $\\{a_{n}\\}$ and $\\{b_{n}\\}$ be two sequences of positive numbers so
that for $0<a_{0}<b_{0}=1$, and for $n\geq 1$ we have $a_{n+1}<b_{n}<a_{n},$
and $a_{n},b_{n}\to 0$ as $n\to\infty$.

For $n\geq 0$ let $I_{n}:=(a_{n},b_{n})$ and
$C_{n}:=(a_{n},b_{n})\times[0,\infty)$ be the “chimney” over the interval
$I_{n}$. We define the domain $W=W(\\{a_{n}\\},\\{b_{n}\\})\subset\mathbb{C}$
as follows:

(3.1) $\displaystyle W=\\{z:\Re z>0,\Im z<0\\}\bigcup_{n=0}^{\infty}C_{n},$

and observe that $W$ is a simply connected domain with a sequence of chimneys
which accumulate to the part of the imaginary axis in the upper half-plane,
i.e., $\\{z\in\mathbb{C}:\Re{z}=0,\Im{z}\geq 0\\}$. Note that $W$ is
completely determined by the sequences $\\{a_{n}\\}_{n\in\mathbb{N}}$ and
$\\{b_{n}\\}_{n\in\mathbb{N}}$

Let $\phi:\mathbb{D}\to W$ be the conformal map with the following properties

(3.2) $\displaystyle\begin{split}\lim_{z\to 0}\phi^{-1}(z)&=-1\\\ \lim_{z\to
1}\phi^{-1}(z)&=1\\\ \lim_{\Im(z)\to-\infty}\phi^{-1}(z)&=-i.\end{split}$

By the discussion in Section 2.6 the path
$\varepsilon\mapsto\psi_{W,\varepsilon}$ is a generalized Teichmüller
geodesic. We will show that this geodesic does not converge to a unique point
in $\partial_{\infty}T(\mathbb{D})$. Moreover, we will describe the limit set
of the geodesic completely and will show that it is in fact homeomorphic to a
one dimensional simplex (an interval). To describe the limit set of
$\\{\psi_{W,\varepsilon}\\}$ we first introduce some notation.

We denote by $z_{n}$ the points on $\mathbb{S}^{1}$ corresponding to the
chimney $C_{n}$ for $n\geq 0$. More specifically, denoting $w=\phi(z)\in W$,
for $n\geq 0$ we let

$\displaystyle z_{n}:=\lim_{\begin{subarray}{c}\Im(w)\to+\infty\\\ \Re(w)\in
I_{n}\end{subarray}}\phi^{-1}(w).$

We also define

$\displaystyle\alpha_{n}=\lim_{w\to
a_{n}}\phi^{-1}(w),\quad\beta_{n}=\lim_{w\to b_{n}}\phi^{-1}(w).$

We will denote by $\gamma_{(x,y)}$ the (unoriented) hyperbolic geodesic in
$\mathbb{D}$ with endpoints $x,y\in\mathbb{S}^{1}$. Moreover, for $n\geq 0$ we
denote by $\gamma_{n}^{+}$ and $\gamma_{n}^{-}$ the two geodesics starting at
$z_{n}$ and ending at $\alpha_{n}$ and $\beta_{n}$, respectively:

$\displaystyle\gamma_{n}^{-}:=\gamma_{(z_{n},\alpha_{n})},\quad\gamma_{n}^{+}:=\gamma_{(z_{n},\beta_{n})}.$

We also denote by $g$ the gedesic from $-1$ to $-i$, i.e.,
$g:=\gamma_{(-1,-i)}$.

Below, as in Section 2.4, we denote by $\tau_{\varepsilon,W}$ the point in
$T(\mathbb{D})$ corresponding to the vertical compression of $W$ by
$\varepsilon>0$, i.e.,

$\tau_{\varepsilon,W}=\left[\left(\frac{1-\varepsilon}{1+\varepsilon}\right)\frac{|\varphi_{W}|}{\varphi_{W}}\right].$

The following result states that the generalized Teichmüller geodesic ray
$\varepsilon\mapsto\tau_{\varepsilon,W},$ does not converge to any point in
Thurston boundary, and, moreover, it describes the limit set of the ray in
Thurston boundary $\partial_{\infty}T(\mathbb{D})$, i.e., all the possible
points in $PLM_{bdd}(\mathbb{D})$ which occur as weak star limits of the
geodesic ray $\varepsilon\mapsto\tau_{\varepsilon,W}.$

We recall that given a hyperbolic geodesic $\gamma=\gamma_{(x,y)}$ in
$\mathbb{D}$ we will denote by $\delta_{\gamma}$ the Dirac mass on $\gamma$.
In particular, if $B=(a,b)\times(c,d)\subset G(\mathbb{D})$ is a box of
geodesics, we will have that

$\displaystyle\delta_{\gamma}(B)=\begin{cases}0,\mbox{ if }\gamma\in B,\\\
1,\mbox{ otherwise.}\end{cases}$

Recall, that $g=\gamma_{(-1,-i)}$ and $\delta_{g}$ is the Dirac mass on the
geodesic connecting $-1$ and $-i$ in $\mathbb{D}$. We will also denote
$\lambda_{W}=\sum_{n=0}^{\infty}(\delta_{\gamma_{n}^{+}}+\delta_{\gamma_{n}^{-}})$.
Thus, $\lambda_{W}$ is a geodesic measured lamination supported on the
geodesics connecting $z_{n}$ to $\alpha_{n}$ and $\beta_{n}$ for
$n\in\mathbb{N}$.

In what follows we use the notation

$\displaystyle A_{n}$ $\displaystyle=\prod_{k=0}^{n}a_{k},\quad\mbox{ and
}\quad B_{n}=\prod_{k=0}^{n}b_{k}=\prod_{k=1}^{n}b_{k}.$

######  Theorem 3.1.

Let $W$ be a domain defined as in (3.1) so that there is constant $0<c<1$ such
that for all $n\in\mathbb{N}$ we have

(3.3) $\displaystyle\max\left({b_{n+1}}/{a_{n}},{a_{n}}/{b_{n}}\right)<c<1$

and

(3.4)
$\displaystyle\max\left\\{\sqrt[n]{a_{n}},\sqrt[n]{b_{n}}\right\\}\underset{n\to\infty}{\to}0.$

Then the limit set $\Lambda$ of the generalized Teichmüller ray
$\varepsilon\mapsto\tau_{\varepsilon,W}$ in $PML_{bdd}(\mathbb{D})$ can be
described as follows:

(3.5) $\displaystyle\Lambda=\\{\llbracket
s\delta_{g}+(2/3)\lambda_{W}\rrbracket:s\in[m,M]\\},$

where

(3.6)
$\displaystyle\begin{split}m&=1+\liminf_{n\to\infty}\frac{\log(A_{n-1}/B_{n})}{\log(1/a_{n})},\quad
M=1+\limsup_{n\to\infty}\frac{\log(A_{n-1}/B_{n})}{\log(1/b_{n})}.\end{split}$

Note that all the geodesic measured laminations $s\delta_{g}+\lambda_{W}$ have
the same support and they only differ by the weight on the geodesic $g$.

###### Proof of Theorem 1.1.

Letting $x_{2n}=b_{n}$ and $x_{2n+1}=a_{n}$ for $n\geq 0$, with $x_{0}=1$ we
see that if $x_{n+1}/x_{n}\to 0$ then (3.3) holds for every $c>0$. Therefore,
for every $\varepsilon\in(0,1)$ there is an $N\in\mathbb{N}$ s.t. for $n\geq
N$ we have $x_{n+2}/x_{n}<\varepsilon$ and hence
$x_{N+2m}\leq\varepsilon^{m}x_{N}$. Thus, if $m$ is so large that $n:=N+2m<3m$
and $x_{N}^{1/n}<2$ then
$x_{n}^{{1}/{n}}\leq\varepsilon^{\frac{m}{N+2m}}x_{N}^{1/n}\leq
2\varepsilon^{1/3}$. Hence $\sqrt[n]{x_{n}}\to 0$ as $n\to\infty$, which
implies (3.4). By the definition of $x_{n}$ above we have that (1.4) s
equivalent to (3.6), which completes the proof. ∎

######  Remark 3.2.

Since
$\log\frac{A_{n-1}}{B_{n}}=\log\frac{A_{n}}{B_{n}}+\log\frac{1}{a_{n}}=\log\frac{A_{n-1}}{B_{n-1}}+\log\frac{1}{b_{n}}$
we have that

(3.7) $\displaystyle
m=2-\limsup_{n\to\infty}\frac{\log(B_{n}/A_{n})}{\log(1/a_{n})},\quad
M=2-\liminf_{n\to\infty}\frac{\log(B_{n}/A_{n})}{\log(1/b_{n+1})}.$

Therefore, since all the limit terms in (3.6) and (3.7) are non-negative, we
have that $1\leq m<M\leq 2$. In particular, every limit set $\Lambda$ as in
(3.5) corresponds to some interval $[m,M]\subset[1,2]$. Below we will provide
specific examples of sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$ showing that
every interval $[m,M]\subset(1,2)$ occurs as a limit set. It is also possible
to have $m=1$ and $M=2$, but we leave such constructions to the reader.

Next we provide some examples of geodesics rays which are either convergent or
divergent at $T_{\infty}(\mathbb{D})$.

######  Example 3.3.

Suppose $b_{0}=1$, $a_{0}=1/2$, $b_{1}=a_{0}/3$, $a_{1}=b_{1}/4$, etc. Thus

$\displaystyle a_{n}=\frac{1}{(2n+2)!},\quad b_{n}=\frac{1}{(2n+1)!}.$

Then $b_{n}/a_{n}=2(n+1)$, ${B_{n}}/{A_{n}}=2^{n+1}(n+1)!=(2n+2)!!$, and
therefore

$\displaystyle\frac{\log\frac{B_{n}}{A_{n}}}{{\log\frac{1}{a_{n}}}}=\frac{\log(2n+2)!!}{\log(2n+2)!}=\frac{\log(2n+2)!!}{\log(2n+2)!!+\log(2n+1)!!}\underset{n\to\infty}{\longrightarrow}\frac{1}{2}.$

Since $\log{a_{n}}/\log{b_{n}}\to 1$, we have that
$\log(B_{n}/A_{n})/\log(1/b_{n+1})\to 1/2$ as well. Therefore $m=M=3/2$ and
the generalized Teichmüller geodesic converges to
$\left\llbracket\frac{3}{2}\delta_{g}+\frac{2}{3}\lambda_{W}\right\rrbracket$.

Generalizing this example slightly, we obtain the following.

######  Example 3.4.

For every $s\in(1,2)$ there is a domain $W$ as above so that

$\displaystyle\tau_{\varepsilon,W}\underset{\varepsilon\to
0}{\longrightarrow}\llbracket s\delta_{g}+(2/3)\lambda_{W}\rrbracket.$

###### Proof.

For all nonnegative integers $x$, we define the _rising factorial_
$(x)^{\bar{n}}$ by

$(x)^{\bar{n}}=x\cdot(x+1)\cdot...\cdot(x+n-1).$

Fix integers $p,q$ and $r$ with $1\leq p,q<r$. Suppose $b_{0}=1$. Let

$\displaystyle a_{n}$
$\displaystyle=\frac{(nr+1)^{\bar{p}}}{(nr+1)^{\bar{r}}}\ b_{n},\text{ for
}n\geq 0,$ $\displaystyle b_{n}$
$\displaystyle=\frac{(nr+1)^{\bar{q}}}{(nr+1)^{\bar{r}}}\ a_{n-1},\text{ for
}n\geq 1.$

Thus $\displaystyle
a_{n}=\prod_{k=1}^{n}\frac{(kr+1)^{\bar{p}}(kr+1)^{\bar{q}}}{(kr+1)^{\bar{r}}(kr+1)^{\bar{r}}}$
and $\displaystyle
b_{n}=\prod_{k=1}^{n}\frac{((k-1)r+1)^{\bar{p}}(kr+1)^{\bar{q}}}{((k-1)r+1)^{\bar{r}}(kr+1)^{\bar{r}}}$.
Then
$\displaystyle\frac{b_{n}}{a_{n}}=\frac{(nr+1)^{\bar{r}}}{(nr+1)^{\bar{p}}}$,
$\displaystyle\frac{B_{n}}{A_{n}}=\prod_{k=1}^{n}\frac{(kr+1)^{\bar{r}}}{(kr+1)^{\bar{p}}}$,
and therefore

$\displaystyle\frac{\log\frac{B_{n}}{A_{n}}}{{\log\frac{1}{a_{n}}}}$
$\displaystyle=\frac{\log\displaystyle\prod_{k=1}^{n}\frac{(kr+1)^{\bar{r}}}{(kr+1)^{\bar{p}}}}{\log\displaystyle\prod_{k=1}^{n}\frac{(kr+1)^{\bar{r}}(kr+1)^{\bar{r}}}{(kr+1)^{\bar{p}}(kr+1)^{\bar{q}}}}=\frac{1}{1+\frac{\log\displaystyle\prod_{k=1}^{n}\frac{(kr+1)^{\bar{r}}}{(kr+1)^{\bar{q}}}}{\log\displaystyle\prod_{k=1}^{n}\frac{(kr+1)^{\bar{r}}}{(kr+1)^{\bar{p}}}}}$
$\displaystyle\underset{n\to\infty}{\longrightarrow}\frac{1}{1+\frac{r-q}{r-p}}=\frac{r-p}{2r-p-q}\in(0,1).$

Since
$\displaystyle\frac{\log{a_{n}}}{\log{b_{n+1}}}\underset{n\to\infty}{\longrightarrow}1$,
we have that
$\displaystyle\frac{\log\frac{B_{n}}{A_{n}}}{\log\frac{1}{b_{n+1}}}\underset{n\to\infty}{\longrightarrow}\frac{r-p}{2r-p-q}$
as well. Therefore $m=M=\displaystyle 2-\frac{r-p}{2r-p-q}\in(1,2)$. By
choosing appropriate $p,q$ and $r$, we obtain all rational $s\in(1,2)$. Hence,
the generalized Teichmüller geodesic converges to $\left\llbracket
s\delta_{g}+\frac{2}{3}\lambda_{W}\right\rrbracket$. ∎

Next we construct the first example of a divergent geodesic.

######  Example 3.5.

Suppose $p>1$, $b_{0}=1$, $a_{0}=a\in(0,1)$ and for $n\geq 1$ we have
$a_{n}=b_{n}^{p}$ and $b_{n+1}=a_{n}^{p}$. Then $a_{n}=a^{p^{2n}}$ and
$b_{n+1}=a^{p^{2n+1}}$. Hence,

$A_{n}=B_{n}^{p}=a^{\frac{p^{2n+2}-1}{p^{2}-1}}=Ca_{n}^{\frac{p^{2}}{p^{2}-1}},$

where $C=a^{-\frac{1}{p^{2}-1}}$. Therefore,

(3.8)
$\displaystyle\begin{split}m&=2-\lim_{n\to\infty}\frac{\log\frac{B_{n}}{A_{n}}}{\log\frac{1}{a_{n}}}=2-\lim_{n\to\infty}\frac{(\frac{1}{p}-1)\log
A_{n}}{\log\frac{1}{a_{n}}}\\\
&=2-\left(1-\frac{1}{p}\right)\frac{p^{2}}{p^{2}-1}=2-\frac{p}{p+1}=1+\frac{1}{p+1}.\end{split}$

Similarly

(3.9)
$\displaystyle\begin{split}M&=2-\lim_{n\to\infty}\frac{\log\frac{B_{n}}{A_{n}}}{\log\frac{1}{b_{n+1}}}=2-\lim_{n\to\infty}\frac{(\frac{1}{p}-1)\log
A_{n}}{\log\frac{1}{a_{n}^{p}}}\\\
&=2-\left(1-\frac{1}{p}\right)\frac{p}{p^{2}-1}=2-\frac{1}{p+1}.\end{split}$

Therefore, if $p>1$ then the limit set $\Lambda$ in this case is given by the
formula (3.5), where
$[m,M]=\left[1+\frac{1}{p+1},2-\frac{1}{p+1}\right]\subset(1,2).$ In
particular, $\Lambda$ is not a point, and geodesic ray
$\varepsilon\mapsto\tau_{\varepsilon,W}$ diverges at infinity.

More generally, we have the following.

######  Example 3.6.

For every nontrivial interval $I=[1+\alpha,1+\beta]\subset(1,2)$ there is a
domain $W$ as above so that the corresponding generalized Teichmüller geodesic
$\varepsilon\mapsto\tau_{\varepsilon,W}$ diverges and its limit set $\Lambda$
in $PML_{bdd}(\mathbb{D})$ is given by

$\displaystyle\Lambda=\\{\llbracket
s\delta_{g}+(2/3)\lambda_{W}\rrbracket:s\in I\\}.$

###### Proof.

Suppose $p,q>1$, $b_{0}=1$, $a_{0}=a\in(0,1)$ and for $n\geq 1$ we have
$a_{n}=b_{n}^{p}$ and $b_{n+1}=a_{n}^{q}$. Then $a_{n}=a^{p^{n}q^{n}}$,
$b_{n}=a^{p^{n}q^{n-1}}$, and therefore

$\displaystyle A_{n}$ $\displaystyle=a^{\frac{(pq)^{n+1}-1}{pq-1}}\asymp
a_{n}^{\frac{pq}{pq-1}},$ $\displaystyle B_{n}$
$\displaystyle=a^{\frac{p((pq)^{n}-1)}{pq-1}}\asymp a_{n}^{\frac{p}{pq-1}}.$

Therefore, since $A_{n}=B_{n}^{p}$ we have

$\displaystyle m$
$\displaystyle=2-\lim_{n\to\infty}\frac{\log\frac{B_{n}}{A_{n}}}{\log\frac{1}{a_{n}}}=2-\lim_{n\to\infty}\frac{(\frac{1}{p}-1)\log
A_{n}}{\log\frac{1}{a_{n}}}$
$\displaystyle=2-\left(1-\frac{1}{p}\right)\frac{pq}{pq-1}=1+\frac{q-1}{pq-1}.$

Similarly

$\displaystyle M$
$\displaystyle=2-\lim_{n\to\infty}\frac{\log\frac{B_{n}}{A_{n}}}{\log\frac{1}{b_{n+1}}}=2-\lim_{n\to\infty}\frac{(\frac{1}{p}-1)\log
A_{n}}{\log\frac{1}{a_{n}^{q}}}$
$\displaystyle=2-\left(1-\frac{1}{p}\right)\frac{p}{pq-1}=2-\frac{p-1}{pq-1}=1+\frac{p(q-1)}{pq-1}.$

Therefore,

$\displaystyle[m,M]=\left[1+\frac{q-1}{pq-1},1+\frac{p(q-1)}{pq-1}\right]\subset(1,2).$

Observe, that if $0<\alpha<\beta<1$ then letting $p=\frac{\beta}{\alpha}$ and
$q=\frac{1-\alpha}{1-\beta}$ we obtain that $m=1+\alpha$ and $M=1+\beta$.
Therefore a limit set of the generalized Teichmüller geodesic can be given by
(3.5) where $[m,M]$ is any interval in $(1,2)$. ∎

##  4\. Proof of Theorem 3.1

To prove that the right hand side is a subset of the left hand side in (3.5)
of Theorem 3.1 it is enough to show that for every $s\in[m,M]$ there is a
sequence $n_{k}=n_{k}(s)$ such that

(4.1)
$\displaystyle\tau_{\varepsilon_{n_{k}},W}\underset{k\to\infty}{\longrightarrow}\llbracket
s\delta_{g}+(2/3)\lambda_{W}\rrbracket\in\Lambda.$

For the opposite inclusion we need to show that every limit point of
$\tau_{\varepsilon,W}$ as $\varepsilon\to 0$ is of the form $\llbracket
s\delta_{g}+(2/3)\lambda_{W}\rrbracket$ for some $s\in[m,M]$.

By Lemma 2.9, (4.1) would follow if we showed that for every $\gamma\in
G(\mathbb{D})$ there is a constant $\eta_{\gamma}>0$ and a sequence
$\\{n_{k}\\}$ so that for $\eta<\eta_{\gamma}$ we have the following:

(4.2)
$\displaystyle\lim_{k\to\infty}\frac{\mathrm{mod}V_{\varepsilon_{n_{k}}}(\phi(\Gamma_{B_{\eta}(\gamma)}))}{\frac{1}{\pi}\log\frac{1}{\varepsilon_{n_{k}}}}=\begin{cases}s,&\mbox{
if }\gamma=g,\\\ \frac{2}{3},&\mbox{ if
}\gamma\in\bigcup_{n=0}^{\infty}\\{\gamma_{n}^{+},\gamma_{n}^{-}\\},\\\
0,&\mbox{otherwise},\end{cases}$

where $s\in[m,M]$. For $\gamma\neq g$ equality (4.2) follows from the proof of
[HŠ21, Theorem 5.1]. Thus, to prove (4.1) it is enough to assume
$\gamma=g=\gamma_{(-1,-i)}$. To simplify the notation for every curve family
$\Gamma$ in $\mathbb{D}$ we denote
$\Gamma^{\varepsilon}=V_{\varepsilon}(\phi(\Gamma)).$ In particular, if $I,J$
are disjoint arcs on the unit circle $\partial\mathbb{D}$ and
$\Gamma_{I,J}=\Gamma(I,J;\mathbb{D})$ is the curve family connecting $I$ to
$J$ in $\mathbb{D}$, we denote

$\displaystyle\Gamma^{\varepsilon}_{I,J}=\Gamma^{\varepsilon}_{I\times
J}=V_{\varepsilon}(\phi(\Gamma_{I,J})).$

The following lemma implies (4.2) in the case of $\gamma=g$ and therefore also
proves Theorem 3.1.

For every pair of disjoint arcs $I$ and $J$ on $\mathbb{S}^{1}$ we define

(4.3)
$\displaystyle\mathbb{M}_{I,J}(\varepsilon):=\frac{\mathrm{mod}\Gamma^{\varepsilon}_{I,J}}{\frac{1}{\pi}\log\frac{1}{\varepsilon}}.$

######  Lemma 4.1.

For every $s\in[m,M]$ there is a sequence $\varepsilon_{n}\in[a_{n+1},a_{n}]$
for which

(4.4) $\displaystyle\lim_{n\to\infty}\mathbb{M}_{I,J}(\varepsilon_{n})=s,$

whenever $I,J$ are disjoint arcs on $\mathbb{S}^{1}$ s. t. $g\in I\times J$
and $1\notin I\cup J$.

Note that the condition on the arcs $I$ and $J$ can be formulated more
concretely as follow. Assuming that $-1\in I$ and $-i\in J$ we have
$J\subset\\{\Im(z)<0\\}$, and $1\notin I$.

Before proceeding, we explain how Lemma 4.1 implies equality (4.2) for
$\gamma=g$. Note that for every $\eta>0$ small enough we can find boxes of
geodesics $B^{\prime}=I^{\prime}\times J^{\prime}$ and
$B^{\prime\prime}=I^{\prime\prime}\times J^{\prime\prime}$ satisfying the
conditions of Lemma 4.1, such that $B^{\prime}\subset B_{\eta}(g)\subset
B^{\prime\prime}$. Therefore, by monotonicity of modulus we have
$\mathrm{mod}\Gamma^{\varepsilon}_{B^{\prime}}\leq\mathrm{mod}\Gamma^{\varepsilon}_{B_{\eta}(g)}\leq\mathrm{mod}\Gamma^{\varepsilon}_{B^{\prime}}$
and since the limit (4.4) is the same for the boxes $B^{\prime}$ and
$B^{\prime\prime}$ it would have to be the same for $B_{\eta}(g)$.

Figure 4.1. Estimating $\mathrm{mod}\Gamma_{k,2}^{\varepsilon}$ from above.

###### Proof of Lemma 4.1.

We start by showing that the limit (4.4) is independent of the arcs $I$ and
$J$. Let

(4.5) $\displaystyle I_{0}$ $\displaystyle=\\{|z|=1\\}\cap\\{\Im(z)>0\\},\quad
J_{0}=\phi^{-1}(\\{0\\}\times(-\infty,-1)).$

######  Proposition 4.2.

If $I$ and $J$ are disjoint arcs of $\mathbb{S}^{1}$ s.t. $g\in I\times J$ and
$1\notin I\cup J$ then for a sequence $\varepsilon_{k}\to 0$ we have

$\displaystyle\lim_{k\to\infty}\mathbb{M}_{I,J}(\varepsilon_{k})=\lim_{k\to\infty}\mathbb{M}_{I_{0},J_{0}}(\varepsilon_{k})$

whenever the limits above exist.

###### Proof of Proposition 4.2.

It is enough to show that there is a constant $C\geq 1$ such that

(4.6)
$\displaystyle|\mathrm{mod}\Gamma^{\varepsilon}(I,J)-\mathrm{mod}\Gamma^{\varepsilon}(I_{0},J_{0})|\leq
C$

for $\varepsilon\to 0$. Let $Q_{1}$ be the first quadrant in the plane, i.e.,
$Q_{1}=\\{z\in\mathbb{C}:\Re z>0,\Im z>0\\}$, and let ${Q_{2},Q_{3},Q_{4}}$
denote the other quadrants ordered counterclockwise. Define

$\displaystyle I_{1}=I\cap(Q_{1}\cup Q_{2}),\quad I_{2}=I\cap Q_{3},\quad
J_{1}=J\cap Q_{3},\quad J_{2}=J\cap Q_{4}.$

Since $\\{1,-i\\}\cap I=\emptyset$ and $\\{-1,1\\}\cap J=\emptyset$ we have
that $I=I_{1}\cup I_{2}$ and $J=J_{1}\cup J_{2}$. Denoting as before
$\Gamma^{\varepsilon}(E,F)=V_{\varepsilon}(\phi(E),\phi(F);W)$, whenever
$E,F\subset\mathbb{S}^{1}$ we have

(4.7)
$\displaystyle\Gamma^{\varepsilon}(I,J)\subset\Gamma^{\varepsilon}(I_{1},J_{1})\cup\Gamma^{\varepsilon}(I_{1},J_{2})\cup\Gamma^{\varepsilon}(I_{2},J_{1})\cup\Gamma^{\varepsilon}(I_{2},J_{2}).$

To simplify the notation, given an arc $I\subset\mathbb{S}$ we let $\tilde{I}$
be the subset of $\partial{W}$ so that $\phi^{-1}(\tilde{I})=I$, and
$\tilde{I}^{\varepsilon}=V_{\varepsilon}(\tilde{I})$, where abusing the
notation slightly by $\phi^{-1}$ we denote the extension of $\phi^{-1}$ to
$\overline{W}$. Note that since $\partial{W}$ is not locally connected at any
point $ia$, with $a>0$, we have that if $0\in I$ we cannot write
$\tilde{I}=\phi(I)$ since $\phi$ does not extend continuously to
$0\in\partial{\mathbb{D}}$. However if $0\notin I$ then $\tilde{I}=\phi(I)$.

Note that $\Gamma^{\varepsilon}(I_{1},J_{2})$ overflows
$\Gamma([0,1],\tilde{J}_{2}^{\varepsilon};W)$. Since
$\tilde{J}_{2}^{\varepsilon}=\phi(J_{2})$ we have that the relative distance
between $[0,1]$ and $\tilde{J}_{2}^{\varepsilon}$ is bounded below and
therefore by Lemma 2.4 there is a $c_{1}>0$ such that
$\mathrm{mod}\Gamma^{\varepsilon}(I_{1},J_{2})\leq\Gamma([0,1],\tilde{J}_{2}^{\varepsilon};W)\leq
c_{1}.$

Since $\phi(I_{2})$ and $\phi(J_{1})$ both belong to the imaginary axis,
relative distance between the $\tilde{I}_{2}^{\varepsilon}$ and
$\tilde{J}_{1}^{\varepsilon}$ does not change with $\epsilon$ and therefore is
bounded below. Thus $\mathrm{mod}\Gamma^{\varepsilon}(I_{2},J_{1})\leq c_{2}$
for some $c_{2}>0$. Finally,
$\Delta(I_{2}^{\varepsilon},J_{2}^{\epsilon})\to\infty$ since
$\mathrm{diam}(I_{2}^{\varepsilon})\to 0$ as $\varepsilon\to 0$.

Thus, the moduli of all the families on the right hand side of (4.7) are
bounded above as $\varepsilon\to 0$, except for
$\mathrm{mod}\Gamma^{\varepsilon}(I_{1},J_{1})$, and as $\varepsilon\to 0$ we
have

$\displaystyle\mathrm{mod}\Gamma^{\varepsilon}(I_{1},J_{1})\leq\mathrm{mod}\Gamma^{\varepsilon}(I,J)\leq\mathrm{mod}\Gamma^{\varepsilon}(I_{1},J_{1})+C_{1}$

Therefore, to prove (4.6) it is enough to show that there is a constant $C\geq
1$ such that

(4.8)
$\displaystyle|\mathrm{mod}\Gamma^{\varepsilon}(I_{1},J_{1})-\mathrm{mod}\Gamma^{\varepsilon}(I_{0},J_{0})|\leq
C.$

Since $I_{1}\subset I_{0}$ we have

$\displaystyle\Gamma^{\varepsilon}(I_{1},J_{1})$
$\displaystyle\subset\Gamma^{\varepsilon}(I_{0},J_{1}\cap
J_{0})\cup\Gamma^{\varepsilon}(I_{0},J_{1}\setminus
J_{0})\subset\Gamma^{\varepsilon}(I_{0},J_{0})\cup\Gamma^{\varepsilon}(I_{0},J_{1}\setminus
J_{0})$

Observe that since $\tilde{J}_{1}\setminus\tilde{J}_{0}$ is either empty or an
interval compactly contained in $\\{0\\}\times(-\infty,0)$, we have that
$\Delta(I_{0},\tilde{J}_{1}^{\varepsilon}\setminus\tilde{J}_{0}^{\varepsilon})$
is independent of $\varepsilon$ and thus is bounded below away from zero.
Hence, by Lemma 2.4 we have

(4.9)
$\displaystyle\mathrm{mod}\Gamma^{\varepsilon}(I_{1},J_{1})\leq\mathrm{mod}\Gamma^{\varepsilon}(I_{0},J_{0})+C.$

On the other hand, we also have the inclusions

$\displaystyle\Gamma^{\varepsilon}(I_{0},J_{0})$
$\displaystyle\subset\Gamma^{\varepsilon}(I_{0}\cap
I_{1},J_{0})\cup\Gamma^{\varepsilon}(I_{0}\setminus I_{1},J_{0})$
$\displaystyle\subset\Gamma^{\varepsilon}(I_{1},J_{0}\cap
J_{1})\cup\Gamma^{\varepsilon}(I_{1},J_{0}\setminus
J_{1})\cup\Gamma^{\varepsilon}(I_{0}\setminus I_{1},J_{0})$
$\displaystyle\subset\Gamma^{\varepsilon}(I_{1},J_{1})\cup\Gamma^{\varepsilon}(I_{1},J_{0}\setminus
J_{1})\cup\Gamma^{\varepsilon}(I_{0}\setminus I_{1},J_{0}).$

Just like above, since $\tilde{J}_{0}\setminus\tilde{J}_{1}$ is compactly
contained in $\\{0\\}\times(-\infty,0)$, we have that
$\Delta(I_{1},\tilde{J}_{0}^{\varepsilon}\setminus\tilde{J}_{1}^{\varepsilon})$
is bounded away from $0$ and hence the modulus of
$\Gamma^{\varepsilon}(I_{1},J_{0}\setminus J_{1})$ is bounded above by a
constant for all $\varepsilon>0$.

Finally, let $N$ be the smallest natural number $n$ so that $z_{n}\in I_{1}$.
From the construction it follows that $\Gamma^{\varepsilon}(I_{0}\setminus
I_{1},J_{0})$ overflows $\Gamma([a_{N},1],\tilde{J}_{0}^{\varepsilon})$. Since
$\Delta([a_{N},1],\tilde{J}_{0}^{\varepsilon})\geq\frac{a_{N}}{1-a_{N}}>0$ for
all $\varepsilon>0$, we again have that
$\mathrm{mod}\Gamma^{\varepsilon}(I_{0}\setminus I_{1},J_{0})$ is bounded
above independently of $\varepsilon$. Therefore, using the inclusions above we
obtain

(4.10)
$\displaystyle\mathrm{mod}\Gamma^{\varepsilon}(I_{0},J_{0})\leq\mathrm{mod}\Gamma^{\varepsilon}(I_{1},J_{1})+C,$

for some constant $C$ and all $\varepsilon>0$. This proves (4.8), which in
turn implies (4.6) and Proposition 4.2 ∎

Continuing with the proof of Lemma 4.1 we observe that
$\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$ is a continuous function of
$\varepsilon$ in $(0,1)$. This follows from Caratheodory’s theorem on
continuous extension of Riemann mapping at the points where $\partial W$ is
locally connected. However, for readers convenience we give a direct proof.
From the definition of $\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$ it is enough to
show that $\mathrm{mod}\Gamma^{\varepsilon}_{I_{0},J_{0}}$ is continuous. Note
that
$\mathrm{mod}\Gamma^{\varepsilon}_{I_{0},J_{0}}=\mathrm{mod}\Gamma(V_{\varepsilon}(\phi(I_{0})),V_{\varepsilon}(\phi(J_{0}));W)=\mathrm{mod}\Gamma(\phi(I_{0}),\\{0\\}\times(-\infty,-\varepsilon);W).$
Therefore if $0<\zeta<\varepsilon$ then
$\Gamma^{\varepsilon}_{I_{0},J_{0}}\subset\Gamma^{\zeta}_{I_{0},J_{0}}$ then

$\mathrm{mod}\Gamma^{\varepsilon}_{I_{0},J_{0}}\leq\mathrm{mod}\Gamma^{\zeta}_{I_{0},J_{0}}\leq\mathrm{mod}\Gamma^{\varepsilon}_{I_{0},J_{0}}+\mathrm{mod}(\Gamma^{\zeta}_{I_{0},J_{0}}\setminus\Gamma^{\varepsilon}_{I_{0},J_{0}}).$

Let $\tau=(\varepsilon-\zeta)/2.$ Then
$\Gamma^{\zeta}_{I_{0},J_{0}}\setminus\Gamma^{\varepsilon}_{I_{0},J_{0}}$
overflows the family of the curves connecting the boundary components of the
annulus $A(-i\frac{\zeta+\varepsilon}{2},\tau,\varepsilon-\tau)$ in $W$.
Therefore, as $\zeta\to\varepsilon$ or, equivalently, if $\tau\to 0$, we have

$\displaystyle|\mathrm{mod}\Gamma^{\zeta}_{I_{0},J_{0}}-\mathrm{mod}\Gamma^{\varepsilon}_{I_{0},J_{0}}|\leq{\pi}\left({\log\frac{\varepsilon-\tau}{\tau}}\right)^{-1}\underset{\tau\to
0}{\longrightarrow}0.$

Hence
${\mathrm{mod}\Gamma^{\varepsilon}_{I_{0},J_{0}}}/({\frac{1}{\pi}\log\frac{1}{\varepsilon}})$
is continuous in $\varepsilon$.

Since $\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$ is continuous, every $s$ which
is bounded between the inferior and superior limits of
$\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$ as $\varepsilon\to 0$ is a
subsequential limit of $\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$. By Proposition
4.2 the same would also hold for $\mathbb{M}_{I,J}(\varepsilon)$, whenever $I$
and $J$ satisfy the conditions of Lemma 4.1. Therefore, Lemma 4.1 follows from
Lemma 5.1 which states that the inferior and superior limits of
$\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$ as $\varepsilon$ approaches $0$ are
equal to $m$ and $M$, respectively. ∎

##  5\. Modulus bounds

By Proposition 4.2 a key step in proving Lemma 4.1 and Theorem 3.1 are bounds
on $\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$, which are established in this
section. To simplify the notation we let

(5.1)
$\displaystyle\Gamma=\phi(\Gamma(I_{0},J_{0};\mathbb{D}))=\Gamma(\phi(I_{0}),\phi(J_{0});W),$

and for every $\varepsilon>0$ we let
$\Gamma^{\varepsilon}=V_{\varepsilon}(\Gamma)$. Since
$V_{\varepsilon}(\phi(I_{0}))=\phi(I_{0})$ we have

(5.2) $\displaystyle\Gamma^{\varepsilon}$
$\displaystyle=\Gamma(\phi(I_{0}),\\{0\\}\times(-\infty,-\varepsilon);W).$

Furthermore, given $z\in\mathbb{C}$, and $0<r<R<\infty$ we will denote

$A(z,r,R)={B(z,R)}\setminus\overline{B(z,r)}.$

To simplify the exposition we denote

(5.3)
$\displaystyle\begin{split}m_{n}&=1+\frac{\log({A_{n-1}}/{B_{n}})}{\log({1}/{a_{n}})}=2-\frac{\log(B_{n}/A_{n})}{\log(1/a_{n})}\\\
M_{n}&=1+\frac{\log({A_{n-1}}/{B_{n}})}{\log({1}/{b_{n}})}=2-\frac{\log(B_{n-1}/A_{n-1})}{\log(1/b_{n})}\end{split}$

Therefore, by Lemma 4.1 and Proposition 4.2, Theorem 3.1 would follow from the
following result.

######  Lemma 5.1.

With the notation above we have the following equalities

(5.4) $\displaystyle\liminf_{\varepsilon\to
0}\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$
$\displaystyle=\liminf_{n\to\infty}m_{n}=m,$ (5.5)
$\displaystyle\limsup_{\varepsilon\to 0}\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$
$\displaystyle=\limsup_{n\to\infty}M_{n}=M.$

The rest of this section is devoted to proving Lemma 5.1.

###  5.1. Lower bounds for $\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$

Figure 5.1. Estimating $\mathrm{mod}\Gamma^{\varepsilon}$ from below.

######  Proposition 5.2.

For $n\geq 1$ we have

(5.6) $\displaystyle\begin{split}\mathbb{M}_{I_{0},J_{0}}(\varepsilon)&\geq
2-\frac{\log{B_{n}}/A_{n}}{\log{1}/{\varepsilon}},\quad\mbox{for}\,\,\varepsilon\in[b_{n+1},a_{n}],\\\
\mathbb{M}_{I_{0},J_{0}}(\varepsilon)&\geq
1+\frac{\log{A_{n-1}}/{B_{n}}}{\log{1}/{\varepsilon}},\quad\mbox{for}\,\,\varepsilon\in[a_{n},b_{n}].\end{split}$

In particular, for $\varepsilon\in[b_{n+1},b_{n}]$ we have

(5.7) $\displaystyle\mathbb{M}_{I_{0},J_{0}}(\varepsilon)\geq m_{n},$

where $m_{n}$ is defined as in (5.3).

###### Proof.

For $k\geq 0$ we denote

$\displaystyle G_{k,1}^{\varepsilon}$
$\displaystyle=\\{\gamma\in\Gamma^{\varepsilon}:\gamma\subset
A(0,a_{k},b_{k})\\},$ $\displaystyle G_{k,2}^{\varepsilon}$
$\displaystyle=\\{\gamma\in\Gamma^{\varepsilon}:\gamma\subset
A(0,b_{k},a_{k-1})\\}.$

Suppose $\varepsilon\in[b_{n+1},a_{n}]$. Denote
$G_{n}^{\varepsilon}=\\{\gamma\in\Gamma^{\varepsilon}|\gamma\subset
A(0,\varepsilon,a_{n})\\}$. Using the motonicity and overflowing properties of
modulus, as well as the fact that the families $G_{n}^{\varepsilon}$ and
$G_{k,i}^{\varepsilon}$ for $k\in\\{0,\ldots,n\\}$ and $i\in\\{1,2\\},$ are
pairwise disjoint, we have

$\displaystyle\mathrm{mod}\Gamma^{\varepsilon}$
$\displaystyle\geq\mathrm{mod}G_{n}^{\varepsilon}+\sum_{k=0}^{n}(\mathrm{mod}G_{k,1}^{\varepsilon}+\mathrm{mod}G_{k,2}^{\varepsilon})$
$\displaystyle\geq\frac{2}{\pi}\log\frac{a_{n}}{\varepsilon}+\frac{1}{\pi}\sum_{k=0}^{n}\log\frac{b_{k}}{a_{k}}+\frac{2}{\pi}\sum_{k=1}^{n}\log\frac{a_{k-1}}{b_{k}}$
$\displaystyle=\frac{2}{\pi}\log\frac{a_{n}}{\varepsilon}+\frac{1}{\pi}\log\frac{B_{n}}{A_{n}}+\frac{2}{\pi}\log\frac{A_{n-1}}{B_{n}}$
$\displaystyle=\frac{2}{\pi}\log\frac{1}{\varepsilon}-\frac{1}{\pi}\log\frac{B_{n}}{A_{n}}.$

Dividing both sides by $\frac{1}{\pi}\log\frac{1}{\varepsilon}$ gives the
first line in (5.6). For $\varepsilon\in[a_{n},b_{n}]$ we similarly estimate

$\displaystyle\mathrm{mod}\Gamma^{\varepsilon}$
$\displaystyle\geq\frac{1}{\pi}\log\frac{b_{n}}{\varepsilon}+\frac{2}{\pi}\log\frac{A_{n-1}}{B_{n}}+\frac{1}{\pi}\log\frac{B_{n-1}}{A_{n-1}}$
$\displaystyle=\frac{1}{\pi}\log\frac{1}{\varepsilon}+\frac{1}{\pi}\log\frac{A_{n-1}}{B_{n}}$

Dividing by $\frac{1}{\pi}\log\frac{1}{\varepsilon}$ gives the second line in
(5.6).

Using (5.6), the fact that $1/(\log(1/\varepsilon))$ is an increasing
function, and the first line in (5.3), we obtain (5.7). ∎

###  5.2. Upper bounds for $\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$

To obtain upper bounds for $\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$ we need an
auxilliary result. As before we let
$\Gamma^{\varepsilon}=V_{\varepsilon}(\Gamma(\phi(I_{0}),\phi(J_{0});W))=\Gamma(\phi(I_{0}),\\{0\\}\times(-\infty,-\varepsilon);W)$
for $\varepsilon\in(0,1)$. Without loss of generality we may assume that
$\gamma(0)$ belongs to the imaginary axis. Thus $\Im(\gamma(0))<-\varepsilon$.
We will denote by $\Gamma_{k,1}^{\varepsilon}$ and
$\Gamma_{k,2}^{\varepsilon}$ the following subfamilies of
$\Gamma^{\varepsilon}$:

$\displaystyle\Gamma_{k,1}^{\varepsilon}$
$\displaystyle=\\{\gamma\in\Gamma^{\varepsilon}:i\gamma(0)\in[a_{k},b_{k}]\\},\quad
k\geq 0,$ $\displaystyle\Gamma_{k,2}^{\varepsilon}$
$\displaystyle=\\{\gamma\in\Gamma^{\varepsilon}:i\gamma(0)\in[b_{k},a_{k-1}]\\},\quad
k\geq 1,$

and let
$\Gamma_{0,2}^{\varepsilon}=\\{\gamma\in\Gamma^{\varepsilon}:i\gamma(0)\in[1,\infty)\\}$.
Note that if $b_{k}<\varepsilon$ then $\Gamma^{\varepsilon}_{k,1}$ is empty,
and if $a_{k-1}<\varepsilon$ then $\Gamma_{k,2}^{\varepsilon}$ is empty.

######  Lemma 5.3.

There is a constant $0<C<\infty$ depending on $c$ such that

(5.8) $\displaystyle\mathrm{mod}\Gamma^{\varepsilon}_{k,1}$
$\displaystyle\leq\frac{1}{\pi}\log\frac{b_{k}}{\max\\{\varepsilon,a_{k}\\}}+C,\,\,\mbox{\rm{for}
}\varepsilon<b_{k},\mbox{ \rm{and} }k\geq 0,$ (5.9)
$\displaystyle\mathrm{mod}\Gamma^{\varepsilon}_{k,2}$
$\displaystyle\leq\frac{2}{\pi}\log\frac{a_{k-1}}{\max\\{\varepsilon,b_{k}\\}}+C,\,\,\mbox{
\rm{for} }\varepsilon<a_{k-1},\mbox{ \rm{and} }k\geq 1,$ (5.10)
$\displaystyle\mathrm{mod}\Gamma^{\varepsilon}_{0,2}$ $\displaystyle\leq 2.$

###### Proof.

Figure 5.2. Estimating $\mathrm{mod}\Gamma_{k,2}^{\varepsilon}$ from above.

To obtain inequality (5.10) observe that if $\gamma\in\Gamma_{0,2}$ then
$\gamma$ contains a subcurve connecting the interval $[0,1]$ and $\\{z\in
W:\mathrm{dist}(z,[0,1])=1\\}\cap\\{\Im(z)<0\\}$. Every such subcurve of
$\gamma$ has length at least $1$ and is contained in the region $\\{z\in
W:\mathrm{dist}(z,[0,1])<1\\}\cap\\{\Im z<0\\}$. The latter region has area
$1+\pi/4<2$ and therefore by the overflowing property we obtain (5.10).

Proof of (5.9). Fix $\varepsilon>0$ and an integer $k\geq 1$ so that
$\varepsilon<a_{k-1}$. If $\varepsilon\geq b_{k}$ let

$\Gamma_{k}^{\prime}=\\{\gamma\in\Gamma_{k,2}^{\varepsilon}:\gamma\subset
A(0,c\varepsilon,a_{k-1}/c)\\}.$

Note that every $\gamma\in\Gamma_{k}^{\prime}$ has a subcurve which separates
the two boundary circles of the annulus $A(0,c\varepsilon,a_{k-1}/c)$ and lies
in the fourth quadrant $Q_{4}$. Hence,

$\displaystyle\mathrm{mod}(\Gamma_{k}^{\prime})\leq\frac{2}{\pi}\log\frac{a_{k-1}/c}{c\varepsilon}=\frac{2}{\pi}\left[\log\frac{a_{k-1}}{\varepsilon}+2\log\frac{1}{c}\right].$

Furthermore, if
$\gamma\in\Gamma_{k,2}^{\varepsilon}\setminus\Gamma_{k}^{\prime}$ then it
connects the two boundary components of either the annulus
$A(0,a_{k-1},a_{k-1}/c)$ or the annulus $A(0,c\varepsilon,\varepsilon)$.
Therefore, since $\gamma\subset\\{\Re z>0\\}$, we have
$\mathrm{mod}(\Gamma_{k,2}^{\varepsilon}\setminus\Gamma_{k}^{\prime})\leq
2\cdot\frac{\pi}{\log(1/c)}.$ Since
$\mathrm{mod}\Gamma_{k,2}^{\varepsilon}\leq\mathrm{mod}\Gamma_{k}^{\prime}+\mathrm{mod}(\Gamma_{k,2}^{\varepsilon}\setminus\Gamma_{k}^{\prime})$,
it follows that (5.9) holds with the constant
$C_{1}=\frac{2}{\pi}\log\frac{1}{c^{2}}+\frac{2\pi}{\log\frac{1}{c}}$ for
$\varepsilon\in[b_{k},a_{k-1}]$.

If $\varepsilon<b_{k}$ then the same argument as above holds by replacing
$\varepsilon$ with $b_{k}$. Therefore we obtains
$\mathrm{mod}\Gamma^{\varepsilon}_{k,2}\leq\frac{1}{\pi}\log\frac{a_{k-1}}{b_{k}}+C_{1}$
thus completing the proof of (5.9).

Proof of (5.8). Fix $\varepsilon>0$ and $k\geq 0$ so that $\varepsilon<b_{k}$.
If $\varepsilon\geq a_{k}$ let

$\Gamma_{k}^{\prime\prime}=\\{\gamma\in\Gamma^{\varepsilon}_{k,1}:\gamma\subset
A(0,c\varepsilon,b_{k}/c)\\},$

and $F_{k}=\Gamma^{\varepsilon}_{k,1}\setminus\Gamma_{k}^{\prime\prime}.$ Just
like above $\mathrm{mod}F_{k}\leq 2\cdot\frac{\pi}{\log(1/c)}.$

To bound from above $\mathrm{mod}\Gamma_{k}^{\prime\prime}$, we let
$\zeta_{k}=(a_{k},0)$ and consider the following subfamilies of
$\Gamma_{k}^{\prime\prime}$:

$\displaystyle\mathscr{F}_{k}^{1}$
$\displaystyle=\\{\gamma\in\Gamma_{k}^{\prime\prime}:\Im(\gamma(1))\leq\varepsilon\\},$
$\displaystyle\mathscr{F}_{k}^{2}$
$\displaystyle=\\{\gamma\in\Gamma_{k}^{\prime\prime}:\Im(\gamma(1))\geq\varepsilon,\gamma\cap
B(\zeta_{k},(1-c)\varepsilon)=\emptyset\\},$
$\displaystyle\mathscr{F}_{k}^{3}$
$\displaystyle=\Gamma_{k}^{\prime\prime}\setminus(\mathscr{F}_{k}^{1}\cup\mathscr{F}_{k}^{2}).$

Figure 5.3. Estimating $\mathrm{mod}\Gamma_{k,1}^{\varepsilon}$ from above.

To estimate $\mathrm{mod}\mathscr{F}_{k}^{1}$, note that for every
$\gamma\in\mathscr{F}_{k}^{1}$ we have either
$\gamma(1)\in[ca_{k},a_{k}]\cup(\\{a_{k}\\}\times[0,\varepsilon])$ or not. In
the former case, since $\Im(\gamma(0))\leq-\varepsilon$ and
$\Im(\gamma(1))\geq 0$, we have that $\mathrm{diam}(\gamma)\geq\varepsilon$.
Thus the modulus of the subfamily of curves $\gamma$ in $\mathscr{F}_{k}^{1}$
with $\gamma(1)\in[ca_{k},a_{k}]\cup(\\{a_{k}\\}\times[0,\varepsilon])$ can be
estimated from above by

$\displaystyle\int_{N_{\varepsilon}([ca_{k},a_{k}]\cup(\\{a_{k}\\}\times[0,\varepsilon])}\varepsilon^{-2}dxdy<8\varepsilon^{2}\cdot\varepsilon^{-2}=8,$

where $N_{r}(E)$ denotes the $r$-neghborhood of a set $E\subset\mathbb{C}$. On
the other hand, if
$\gamma(1)\notin[ca_{k},a_{k}]\cup(\\{a_{k}\\}\times[0,\varepsilon])$ then
either $\gamma$ intersects the horizontal line $\\{\Im(z)=b_{k}\\}$ or not. If
$\gamma\cap\\{\Im(z)=b_{k}\\}\neq\emptyset$ then $\gamma$ contains a subcurve
which connects the horizontal sides of the square $[0,b_{k}]\times[0,b_{k}]$.
If $\gamma\cap\\{\Im(z)=b_{k}\\}=\emptyset$ then $\gamma$ contains a subcurve
which connects the vertical sides of the rectangle
$[0,b_{k}]\times[-c^{-1}b_{k},b_{k}]$. Therefore, using the overflowing
property we obtain:

$\displaystyle\mathrm{mod}\mathscr{F}_{k}^{1}\leq
8+1+\frac{b_{k}-(-b_{k}/c)}{b_{k}}=10+\frac{1}{c}.$

To estimate the modulus of $\mathscr{F}_{k}^{2}$, we first pick an arbitrary
curve $\gamma\in\mathscr{F}_{k}^{2}$. Note, that since $b_{k+1}/a_{k}<c$,
$b_{k}/a_{k-1}<c$, and $\gamma\subset A(0,c\varepsilon,c^{-1}b_{k})$ we have
that

$\displaystyle\gamma(1)\in(\\{a_{k}\\}\cup\\{b_{k}\\})\times(\varepsilon,\infty).$

In particular, $\Re(\gamma(1))\in\\{a_{k},b_{k}\\}$. If $\Re(\gamma(1))=b_{k}$
then like above we have two cases: either
$\gamma\subset[0,b_{k}]\times[-b_{k}/c,b_{k}]$ or not. As above in the former
case $\gamma$ connects the vertical sides of
$[0,b_{k}]\times[-b_{k}/c,b_{k}]$, and in the latter case it has a subcurve
connecting the horizontal sides of the square $[0,b_{k}]\times[0,b_{k}]$.
Therefore the modulus of the family
$\\{\gamma\in\mathscr{F}_{k}^{2}:\Re(\gamma(1))=b_{k}\\}$ is bounded above by
$\frac{b_{k}/c+b_{k}}{b_{k}}+1=c^{-1}+2$.

Next, suppose $\gamma\in\mathscr{F}_{k}^{2}$ is such that
$\Re(\gamma(1))=a_{k}$. Recall, that $\zeta_{k}=(a_{k},0)\in\mathbb{R}$ and
consider the annulus

$R_{k}=A(\zeta_{k},(1-c)\varepsilon,c^{-1}b_{k}).$

Since $\gamma\cap B(\zeta_{k},(1-c)\varepsilon)=\emptyset$ it follows that
$\gamma$ intersects the vertical interval
$\\{a_{k}\\}\times[-b_{k}/c,-(1-c)\varepsilon]$ and
$\gamma\cap\\{\Re{z}>a_{k}\\}$ is contained in $R_{k}$. Therefore,
$\mathscr{F}_{k}^{2}$ overflows the family of curves in the semi-annulus
$R_{k}\cap\\{\Re z>a_{k}\\}$ which separate the two boundary components of
${R}_{k}$. The modulus of the latter family is

$\frac{1}{\pi}\log\frac{b_{k}/c}{(1-c)\varepsilon}=\frac{1}{\pi}\log\frac{b_{k}}{\varepsilon}+\frac{1}{\pi}\log\frac{1}{c(1-c)}.$

Combining the two cases above we obtain

$\displaystyle\mathrm{mod}\mathscr{F}_{k}^{2}\leq\frac{1}{\pi}\log\frac{b_{k}}{\varepsilon}+\frac{1}{\pi}\log\left(\frac{1}{c(1-c)}\right)+\frac{1}{c}+2.$

Finally, $\mathscr{F}_{k}^{3}$ overflows the family of curves connecting the
boundary components of the annulus
$A(\zeta_{k},(1-c)\varepsilon,\varepsilon)$, which has modulus
$\frac{2\pi}{\log\frac{1}{1-c}}$.

Since
$\mathrm{mod}\Gamma_{k,1}^{\varepsilon}\leq\mathrm{mod}(\mathscr{F}_{k}^{1}\cup\mathscr{F}_{k}^{2}\cup\mathscr{F}_{k}^{3}\cup
F_{k})$, combining the estimates above we obtain (5.8) with
$C_{2}=12+\frac{2}{c}+\frac{2\pi}{\log\frac{1}{c}}+\frac{2\pi}{\log\frac{1}{1-c}}+\frac{1}{\pi}\left[\log\frac{1}{c}+\log\frac{1}{1-c}\right]$.
Therefore Lemma 5.3 holds with $C=\max(C_{1},C_{2})$. ∎

We are now ready to estimate $\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$ from
above.

######  Proposition 5.4.

For $n\geq 1$ we have

(5.11) $\displaystyle\begin{split}\mathbb{M}_{I_{0},J_{0}}(\varepsilon)&\leq
2-\frac{\log{B_{n}}/{A_{n}}}{\log{1}/{\varepsilon}}+o(1),\quad\mbox{ for
}\varepsilon\in[b_{n+1},a_{n}]\\\ \mathbb{M}_{I_{0},J_{0}}(\varepsilon)&\leq
1+\frac{\log{A_{n-1}}/{B_{n}}}{\log{1}/{\varepsilon}}+o(1),\quad\mbox{ for
}\varepsilon\in[a_{n},b_{n}].\end{split}$

In particular, for $\varepsilon\in[a_{n},a_{n-1}]$ we have

(5.12) $\displaystyle\mathbb{M}_{I_{0},J_{0}}(\varepsilon)\leq M_{n}+o(1),$

where $M_{n}$ is defined in (5.3).

###### Proof.

Suppose $\varepsilon\in[b_{n+1},a_{n}]$, for some $n\geq 1$. Then
$\Gamma_{n+1,2}^{\varepsilon}=\\{\gamma\in\Gamma^{\varepsilon}:i\gamma(0)\in[\varepsilon,a_{n}]\\}$.
Therefore, by Lemma 5.3 there is a constant $C$ (not the same though as in
estimates (5.8) and (5.9)) so that for $n$ large enough we have

$\displaystyle\mathrm{mod}\Gamma^{\varepsilon}$
$\displaystyle\leq\mathrm{mod}\Gamma^{\varepsilon}_{n+1,2}+\sum_{k=0}^{n}\left(\mathrm{mod}\Gamma_{k,1}^{\varepsilon}+\mathrm{mod}\Gamma_{k,2}^{\varepsilon}\right)$
$\displaystyle\leq\frac{2}{\pi}\log\frac{a_{n}}{\varepsilon}+\frac{1}{\pi}\sum_{k=0}^{n}\log\frac{b_{k}}{a_{k}}+\frac{2}{\pi}\sum_{k=1}^{n}\log\frac{a_{k-1}}{b_{k}}+Cn$
$\displaystyle\leq\frac{2}{\pi}\log\frac{1}{\varepsilon}-\frac{1}{\pi}\log\frac{B_{n}}{A_{n}}+Cn$

Since $\varepsilon\in[b_{n+1},a_{n}]$ we obtain

$\displaystyle\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$ $\displaystyle\leq
2-\frac{\log(B_{n}/A_{n})}{\log(1/\varepsilon)}+\frac{Cn}{\log(1/a_{n})}\leq
2-\frac{\log(B_{n}/A_{n})}{\log(1/\varepsilon)}-\frac{C}{\log(\sqrt[n]{a_{n}})}$

Since $\sqrt[n]{a_{n}}\to 0$ as $n\to 0$, we obtain (5.11) for
$\varepsilon\in[b_{n+1},a_{n}]$.

Suppose $\varepsilon\in[a_{n},b_{n}]$. Then
$\Gamma_{n,1}^{\varepsilon}=\\{\gamma\in\Gamma^{\varepsilon}:i\gamma(0)\in[\varepsilon,b_{n}]\\}$.
Just like above, by estimate (5.9) of Lemma 5.3 there is a (possibly new)
constant $C$ such that

$\displaystyle\mathrm{mod}\Gamma^{\varepsilon}$
$\displaystyle\leq\mathrm{mod}\Gamma_{n,1}^{\varepsilon}+\mathrm{mod}\Gamma_{n,2}^{\varepsilon}+\sum_{k=0}^{n-1}\left(\mathrm{mod}\Gamma_{k,1}^{\varepsilon}+\mathrm{mod}\Gamma_{k,2}^{\varepsilon}\right)$
$\displaystyle\leq\frac{1}{\pi}\log\frac{b_{n}}{\varepsilon}+\frac{1}{\pi}\sum_{k=0}^{n-1}\log\frac{b_{k}}{a_{k}}+\frac{2}{\pi}\sum_{k=0}^{n}\log\frac{a_{k-1}}{b_{k}}+Cn$
$\displaystyle=\frac{1}{\pi}\log\frac{1}{\varepsilon}+\frac{1}{\pi}\log\frac{B_{n}}{A_{n-1}}+\frac{2}{\pi}\log\frac{A_{n-1}}{B_{n}}+Cn$
$\displaystyle=\frac{1}{\pi}\log\frac{1}{\varepsilon}+\frac{1}{\pi}\log\frac{A_{n-1}}{B_{n}}+Cn.$

Dividing both sides by $\frac{1}{\pi}\log(1/\varepsilon)$ and using the fact
that $\varepsilon\leq b_{n}$ we obtain

$\displaystyle\mathbb{M}_{I_{0},J_{0}}(\varepsilon)$ $\displaystyle\leq
1+\frac{\log(A_{n-1}/B_{n})}{\log(1/\varepsilon)}+\frac{Cn}{\log(1/\varepsilon)}\leq
1+\frac{\log(A_{n-1}/B_{n})}{\log(1/\varepsilon)}-\frac{C}{\log(\sqrt[n]{b_{n}})}.$

Since $\sqrt[n]{b_{n}}\to 0$ as $n\to 0$ we obtain (5.11) for
$\varepsilon\in[a_{n},b_{n}]$.

Since the right hand side in the first inequality in (5.11) is decreasing in
$\varepsilon$ we obtain that for $\varepsilon\in[b_{n},a_{n-1}]$ we have
$\mathbb{M}_{I_{0},J_{0}}(\varepsilon)\leq M_{n}+o(1)$. Similarly, the right
hand side in the second inequality in (5.11) is increasing in $\varepsilon$
and hence the maximum in the interval $[a_{n},b_{n}]$ is attained for
$\varepsilon=b_{n}$. This gives (5.12) in $[a_{n},b_{n}]$, thus completing the
proof. ∎

###  5.3. Completing the proof of Lemma 5.1

By Proposition 5.2 we have that the left hand side in (5.4) is greater than or
equal to $m$. To show the opposite inequality, let $n_{k}$ be such that
$\lim_{k\to\infty}m_{n_{k}}=m$. Then letting $\varepsilon_{k}=a_{n_{k}}$ in
the second inequality in (5.11) we obtain

(5.13) $\displaystyle\mathbb{M}_{I_{0},J_{0}}(\varepsilon_{k})\leq
1+\frac{\log{A_{n_{k}-1}}/{B_{n_{k}}}}{\log{1}/{a_{n_{k}}}}+o(1)\underset{k\to\infty}{\longrightarrow}m,$

which proves (5.4).

Similarly, Proposition 5.4 implies that

$\limsup_{\varepsilon\to
0}\mathbb{M}_{I_{0},J_{0}}(\varepsilon)\leq\limsup_{n\to\infty}(M_{n}+o(1))=M,$

and therefore we only need to show the opposite inequality. Let $n_{l}$ be
such that $\lim_{l\to\infty}M_{n_{l}}=M$. Then letting
$\varepsilon_{l}=b_{n_{l}}$ in the second inequality in (5.6) we obtain

(5.14) $\displaystyle\mathbb{M}_{I_{0},J_{0}}(\varepsilon_{l})$
$\displaystyle\geq
1+\frac{\log{A_{n_{l}-1}}/{B_{n_{l}}}}{\log{1}/{b_{n_{l}}}}\underset{l\to\infty}{\longrightarrow}M.$

This completes the proof of inequality (5.5) and, as explained in the
beginning of this section, also proves Theorem 3.1.

##  6\. Limit sets of higher dimensions

In this section, for every $k\geq 2$ we construct a domain with chimneys
$W_{k}$ such that the generalized Teichmüller geodesic ray
$\varepsilon\mapsto\tau_{W_{k},\varepsilon}$ has a limit set $\Lambda$ in
Thurston boundary $T_{\infty}(\mathbb{D})$ which is a $k$ simplex, i.e., is
homeomorphic to $[0,1]^{k}$. We will give the detailed construction for $k=2$
and explain how the case of general $k\geq 2$ can be constructed in a similar
way.

###  6.1. Limit sets of dimension $2$

Let $p,q\in(1,\infty)$ and $a\in(0,1)$. Let $b_{0}=1$, $a_{0}=a$,
$b_{1}=a^{p}$. For $n\geq 1$, let $a_{n}=b_{n}^{p}$, and $b_{n}=a_{n-1}^{p}$.
Therefore, for $n\geq 1$ we have

$\displaystyle a_{n}=a_{n-1}^{p^{2}}=a^{p^{2n}},\quad b_{n}=a^{p^{2n-1}}.$

Also, let $d_{0}=2$, $c_{0}=3-a$, $d_{1}=3-a^{q}$. For $n\geq 1$, let
$c_{n}=3-(3-d_{n})^{q}$, and $d_{n}=3-(3-c_{n-1})^{q}$. Therefore for $n\geq
1$ we have

$\displaystyle c_{n}=3-a^{q^{2n}},\quad d_{n}=3-a^{q^{2n-1}}.$

Below it will be convenient to use the following notation

(6.1) $\displaystyle u_{n}=3-c_{n}=a^{q^{2n}},\quad
v_{n}=3-d_{n}=a^{q^{2n-1}}.$

Moreover, we let

(6.2) $\displaystyle U_{n}=\prod_{i=0}^{n}u_{i},\quad
V_{n}=\prod_{i=0}^{n}v_{i}.$

Define the chimneys $C_{n}=(a_{n},b_{n})\times\mathbb{R}$ and
$C_{n}^{\prime}=(c_{n},d_{n})\times\mathbb{R}$ and the domain $W=W_{p,q}$ by

(6.3) $\displaystyle
W_{p,q}=\\{z:\Im{z}<0,0<\Re(z)<3\\}\cup\bigcup_{i=1}^{\infty}C_{n}\cup
C_{n}^{\prime}.$

We denote by $\psi:\mathbb{D}\to W$ the conformal map such that

(6.4) $\displaystyle\begin{split}\lim_{z\to 0}\psi^{-1}(z)&=-1,\\\ \lim_{z\to
3}\psi^{-1}(z)&=1,\\\ \lim_{\Im(z)\to-\infty}\psi^{-1}(z)&=-i.\end{split}$

We denote by $a_{n}^{\prime},b^{\prime}_{n},c^{\prime}_{n}$ and
$d^{\prime}_{n}$ the preimages under $\psi$ of $a_{n},b_{n},c_{n}$ and
$d_{n}$, respectively. Also, we define $z_{n}$ and $z^{\prime}_{n}$ as
follows:

$\displaystyle\lim_{\begin{subarray}{c}\Im(z)\to+\infty\\\ z\in
C_{n}\end{subarray}}\psi^{-1}(z)$ $\displaystyle=z_{n},$
$\displaystyle\lim_{\begin{subarray}{c}\Im(z)\to+\infty\\\ z\in
C^{\prime}_{n}\end{subarray}}\psi^{-1}(z)$ $\displaystyle=z^{\prime}_{n}.$

As before we denote by $\gamma_{(x,y)}$ the hyperbolic geodesic in
$\mathbb{D}$ with endpoints $x,y\in\mathbb{S}^{1}$. Moreover, for $n\geq 0$ we
denote by $\gamma_{n}^{+}$ and $\gamma_{n}^{-}$ the two geodesics starting at
$z_{n}$ and ending at $a^{\prime}_{n}$ and $b^{\prime}_{n}$, respectively.
Similarly, for $n\leq 0$ we denote by $\tilde{\gamma}_{n}^{+}$ and
$\tilde{\gamma}_{n}^{-}$ the geodesics starting at $z^{\prime}_{n}$ and ending
at $c^{\prime}_{n}$ and $d^{\prime}_{n}$, respectively. With this notation in
hand we define the measured lamination $\lambda_{W}$ by

(6.5)
$\displaystyle\lambda_{W}=\sum_{n=0}^{\infty}(\delta_{\gamma_{i}^{+}}+\delta_{\gamma_{i}^{-}})+\sum_{n=0}^{\infty}(\delta_{\tilde{\gamma}_{i}^{+}}+\delta_{\tilde{\gamma}_{i}^{-}})$

Let $g_{1}=\gamma_{(-1,-i)}$ and $g_{2}=\gamma_{(1,-i)}$.

Theorem 6.1 gives an example of a geodesic ray in $T(\mathbb{D})$ with the
limit set in Thurston boundary being homeomorphic to a square.

######  Theorem 6.1.

Suppose $p,q>1$ are such that $\log p/\log q$ is an irrational number. If the
domain $W$ is defined as in (6.3) then the generalized Teichmüller geodesic
$\varepsilon\mapsto\tau_{\varepsilon,W}$ diverges and the accumulation set
$\Lambda$ in $\partial_{\infty}T(\mathbb{D})$ can be described as follows:

(6.6)
$\displaystyle\Lambda=\\{\llbracket\lambda_{s,t}\rrbracket:(s,t)\in[m_{p},M_{p}]\times[m_{q},M_{q}]\\},$

where $\lambda_{s,t}=s\delta_{g_{1}}+t\delta_{g_{2}}+\frac{2}{3}\lambda_{W}$,
$m_{\ell}=1+\frac{1}{1+\ell}$ and $M_{\ell}=2-\frac{1}{1+\ell}$.

Observe that all the measured laminations $\lambda_{s,t}$ are supported on the
same geodesic lamination, which we will denote by $\Sigma$. Thus,

$\Sigma=g_{1}\cup
g_{2}\bigcup_{n=0}^{\infty}\gamma_{n}^{+}\cup\gamma_{n}^{-}\cup\tilde{\gamma}_{n}^{+}\cup\tilde{\gamma}_{n}^{-}.$

###### Proof.

We first show the inclusion $\supset$ in (6.21). For this we will show that
for every $(s,t)\in[m_{p},M_{p}]\times[m_{q},M_{q}]$ there is a sequence
$\varepsilon_{n_{k}}\to 0$ such that

(6.7)
$\displaystyle\tau_{\varepsilon_{n_{k}},W}\underset{k\to\infty}{\longrightarrow}\llbracket\lambda_{s,t}\rrbracket.$

To formulate (6.7) in terms of the asymptotics of moduli of curve families we
let $I_{0}=\partial(W\cap\\{\Im z>0\\})$, $J_{0}=\\{0\\}\times(-\infty,-1)$,
$J_{1}=\\{3\\}\times(-\infty,-1)$ and denote

$\displaystyle\Gamma^{\varepsilon}$
$\displaystyle=V_{\varepsilon}(\Gamma(I_{0},J_{0};W))=\Gamma(I_{0},\\{0\\}\times(-\infty,-\varepsilon);W)$
$\displaystyle\tilde{\Gamma}^{\varepsilon}$
$\displaystyle=V_{\varepsilon}(\Gamma(I_{0},J_{1};W))=\Gamma(I_{0},\\{3\\}\times(-\infty,-\varepsilon);W).$

Just like in Section 4 to prove (6.7) it is enough to show that if $\Sigma\cap
I\times J=\emptyset$ then
$\lim_{k\to\infty}\mathbb{M}_{I,J}(\varepsilon_{n_{k}})=0$, while if
$\\#(\Sigma\cap I\times J)=1$ then

(6.8)
$\displaystyle\lim_{k\to\infty}\mathbb{M}_{I,J}(\varepsilon_{n_{k}})=\begin{cases}s,&\mbox{
if }\,\,\Sigma\cap I\times J=g_{1},\\\ t,&\mbox{ if }\,\,\Sigma\cap I\times
J=g_{2},\\\ \frac{2}{3},&\,\,\mbox{otherwise}.\end{cases}$

The third case in (6.8) follows just like in [HŠ21]. Moreover, arguing like in
Section 4 we see that instead of general boxes $I\times J$ it is enough to
consider the limits of $\mathbb{M}_{I_{0},J_{0}}(\varepsilon_{n_{k}})$ and
$\mathbb{M}_{I_{0},J_{1}}(\varepsilon_{n_{k}})$. Therefore (6.7) follows from
the following lemma.

######  Lemma 6.2.

For every $(s,t)\in[m_{p},M_{p}]\times[m_{q},M_{q}]$ there is a sequence
$\varepsilon_{n_{k}}\to 0$ such that

(6.9)
$\displaystyle\begin{split}\lim_{k\to\infty}\mathbb{M}_{I_{0},J_{0}}(\varepsilon_{n_{k}})&=s,\\\
\lim_{k\to\infty}\mathbb{M}_{I_{0},J_{1}}(\varepsilon_{n_{k}})&=t.\end{split}$

###### Proof of Lemma 6.2.

From Propositions 5.2 and 5.4 for $n\geq 1$ we have

(6.10)
$\displaystyle\begin{split}0&\leq\mathbb{M}_{I_{0},J_{0}}(\varepsilon)-\left[2-\frac{\log{B_{n}}/{A_{n}}}{\log{1}/{\varepsilon}}\right]\leq
o(1),\quad\mbox{ for }\varepsilon\in[b_{n+1},a_{n}]\\\
0&\leq\mathbb{M}_{I_{0},J_{0}}(\varepsilon)-\left[1+\frac{\log{A_{n-1}}/{B_{n}}}{\log{1}/{\varepsilon}}\right]\leq
o(1),\quad\mbox{ for }\varepsilon\in[a_{n},b_{n}].\end{split}$

For $\alpha\in[1,p^{2}]$ we consider $\varepsilon_{n}=b_{n}^{\alpha}$. From
the definitions we have $\varepsilon_{n}\in[a_{n},b_{n}]$ for $\alpha\in[1,p]$
and $\varepsilon_{n}\in[b_{n+1},a_{n}]$ for $\alpha\in[p,p^{2}]$. Therefore,

$\displaystyle\lim_{n\to\infty}\frac{\mathrm{mod}\Gamma^{\varepsilon_{n}}}{\frac{1}{\pi}\log\frac{1}{\varepsilon_{n}}}$
$\displaystyle=\begin{cases}\displaystyle{\lim_{n\to\infty}}1+\frac{\log
A_{n-1}/B_{n}}{\alpha\log\frac{1}{b_{n}}},&\mbox{ for }\alpha\in[1,p]\\\
\displaystyle{\lim_{n\to\infty}}2-\frac{\log{B_{n}}/{A_{n}}}{\alpha\log\frac{1}{b_{n}}},&\mbox{
for }\alpha\in[p,p^{2}].\end{cases}$

From equations (3.8) and (3.9) in Example 3.5, it follows that
$\displaystyle{\lim_{n\to\infty}}\mathbb{M}_{I_{0},J_{0}}(b_{n}^{\alpha})=\Phi_{p}(\alpha)$,
where

(6.11)
$\displaystyle\Phi_{p}(\alpha)=\begin{cases}1+\frac{p}{\alpha(p+1)},&\mbox{
for }\alpha\in[1,p]\\\ 2-\frac{p^{2}}{\alpha(p+1)},&\mbox{ for
}\alpha\in[p,p^{2}].\end{cases}$

Observe that $\Phi_{p}(\alpha)$ decreases from $\Phi_{p}(1)=M_{p}$ to
$\Phi_{p}(p)=m_{p}$ on $\alpha\in[1,p]$ and then increases from $m_{p}$ to
$\Phi_{p}(p^{2})=\Phi_{p}(1)=M_{p}$ on $\alpha\in[p,p^{2}]$. Therefore, for
every $s\in[m_{p},M_{p}]$ we can find $\alpha\in[1,p]$ (or
$\alpha\in[p,p^{2}]$) so that $\Phi_{p}(\alpha)=s$, and with this choice of
$\alpha$ we have that $\mathbb{M}_{I_{0},J_{0}}(b_{n}^{\alpha})\to s$ as
$n\to\infty$. In particular, for every subsequence $\varepsilon_{n_{k}}$ of
$\varepsilon_{n}=b_{n}^{\alpha}$ the first equation in (6.9) holds, but not
necessarily the second equation.

Next, observe that for every $n$ there is a unique $m=m_{n}\in\mathbb{N}$ such
that

$\varepsilon_{n}=b_{n}^{\alpha}=a^{\alpha
p^{2n-1}}\in(v_{m+1},v_{m}]=(a^{q^{2m+1}},a^{q^{2m-1}}].$

Therefore, $\exists\,\,\beta_{n}\in[0,1)$ such that
$\varepsilon_{n}=v_{m}^{q^{2\beta_{n}}}=a^{q^{2m-1+2\beta_{n}}}$, or

$p^{2n-1+\log_{p}\alpha}=q^{2m-1+2\beta_{n}}.$

Solving for $\beta_{n}$ we obtain

$\displaystyle\beta_{n}$ $\displaystyle=\left(2n-1+\frac{\log\alpha}{\log
p}\right)\frac{\log p}{2\log q}+\frac{1}{2}-m=\theta n+\sigma-m,$

where $\theta:=\log_{q}p$ and
$\sigma:=\frac{1}{2}(1+\log_{q}\frac{\alpha}{p})$. Hence

(6.12) $\displaystyle\beta_{n}\equiv\theta n+\sigma\,\,(\mathrm{mod}\,1)\equiv
T^{\circ n}_{\theta}(\sigma)$

is the $n$-th iterate of $\sigma$ under the map $T_{\theta}:x\mapsto
x+\theta\,\,(\mathrm{mod}\,1)$. Since $\theta\in\mathbb{R}\setminus\mathbb{Q}$
the map $T_{\theta}$ is an irrational rotation, and every orbit
$\\{T_{\theta}^{\circ n}(\sigma)\\}_{n\in\mathbb{N}}$ is dense in $[0,1]$, see
for instance [BS02]. Therefore for every $\beta\in[0,1]$ there is a sequence
$n_{k}$ such that $\beta_{n_{k}}\underset{k\to\infty}{\longrightarrow}\beta$.
Moreover, $\beta_{n_{k}}$ can be chosen to be a monotone sequence. We claim
that

(6.13)
$\displaystyle\lim_{k\to\infty}\mathbb{M}_{I_{0},J_{1}}(\varepsilon_{n_{k}})=\Phi_{q}(q^{2\beta}).$

As in the proofs of Lemma 5.1, Propositions 5.2 and 5.4, we have the following
estimates:

(6.14)
$\displaystyle\begin{split}0&\leq\mathbb{M}_{I_{0},J_{1}}(\varepsilon)-\left[2-\frac{\log{V_{n}}/{U_{n}}}{\log{1}/{\varepsilon}}\right]\leq
o(1),\quad\mbox{ for }\varepsilon\in[v_{n+1},u_{n}]\\\
0&\leq\mathbb{M}_{I_{0},J_{1}}(\varepsilon)-\left[1+\frac{\log{U_{n-1}}/{V_{n}}}{\log{1}/{\varepsilon}}\right]\leq
o(1),\quad\mbox{ for }\varepsilon\in[u_{n},v_{n}].\end{split}$

Therefore, since
$\varepsilon_{n_{k}}=(a^{q^{2m-1}})^{q^{2\beta_{n_{k}}}}=(v_{m})^{q^{2\beta_{n_{k}}}}$
we have

(6.15)
$\displaystyle\begin{split}0&\leq\mathbb{M}_{I_{0},J_{1}}(\varepsilon_{n_{k}})-\left[2-\frac{\log{V_{m}}/{U_{m}}}{q^{2\beta_{n_{k}}}\log{1}/{v_{m}}}\right]\leq
o(1),\quad\mbox{ for }\varepsilon_{n_{k}}\in[v_{m}^{q^{2}},v_{m}^{q}]\\\
0&\leq\mathbb{M}_{I_{0},J_{1}}(\varepsilon_{n_{k}})-\left[1+\frac{\log{U_{m-1}}/{V_{m}}}{q^{2\beta_{n_{k}}}\log{1}/{v_{m}}}\right]\leq
o(1),\quad\mbox{ for }\varepsilon_{n_{k}}\in[v_{m}^{q},v_{m}].\end{split}$

Note that
$\varepsilon_{n_{k}}=(v_{m})^{q^{2\beta_{n_{k}}}}\in[v_{m}^{q},v_{m}]$ if and
only if $\beta_{n_{k}}\in[0,1/2]$. Therefore, for any $\beta\in[0,1/2]$
choosing $\beta_{n_{k}}\in[0,1/2]$ so that $\beta_{n_{k}}\to\beta$ we obtain

(6.16)
$\displaystyle\begin{split}\lim_{k\to\infty}\mathbb{M}_{I_{0},J_{1}}(\varepsilon_{n_{k}})&=\lim_{k\to\infty}1+\frac{\log{U_{m-1}}/{V_{m}}}{q^{2\beta_{n_{k}}}\log{1}/{v_{m}}}\\\
&=1+\frac{1}{q^{2\beta}}\lim_{m\to\infty}\frac{(1/q-1)\log V_{m}}{\log
1/v_{m}}.\\\
&=1+\frac{1}{q^{2\beta}}\left(\frac{1}{q}-1\right)\lim_{m\to\infty}\frac{q^{2m+2}-1}{(q^{2}-1)q}\cdot\frac{1}{-q^{2m-1}}\\\
&=1+\frac{1}{q^{2\beta}}\frac{q}{q+1}=\Phi_{q}(q^{2\beta}).\ \end{split}$

Similarly,
$\varepsilon_{n_{k}}=(v_{m})^{q^{2\beta_{n_{k}}}}\in[v_{m}^{q^{2}},v_{m}^{q}]$
if and only if $\beta_{n_{k}}\in[1/2,1]$. Therefore, for any $\beta\in[1/2,1]$
choosing $\beta_{n_{k}}\in[1/2,1]$ so that $\beta_{n_{k}}\to\beta$ we obtain

$\displaystyle\lim_{k\to\infty}\mathbb{M}_{I_{0},J_{1}}(\varepsilon_{n_{k}})$
$\displaystyle=\lim_{k\to\infty}2-\frac{\log{V_{m}}/{U_{m}}}{q^{2\beta_{n_{k}}}\log{1}/{v_{m}}}=2-\frac{1}{q^{2\beta}}\frac{q^{2}}{q+1}=\Phi_{q}(q^{2\beta}),$

which proves (6.13).

Note that $\Phi_{q}(q^{2\beta})$ continously decreases from
$\Phi_{q}(1)=M_{q}=1+\frac{q}{q+1}$ to $\Phi_{q}(q)=m_{q}=1+\frac{1}{q+1}$ as
$\beta\in[0,1/2]$. Similarly, $\Phi_{q}(q^{2\beta})$ increases from $m_{q}$ to
$M_{q}$ in $\beta\in[1/2,1]$. Therefore, for every $t\in[m_{q},M_{q}]$ there
is a $\beta\in[0,1/2]$ (or $\beta\in[1/2,1]$) such that
$\Phi_{q}(q^{2\beta})=t$. For this choice of $\beta$ choosing $n_{k}$ so that
(6.13) holds gives the second line in (6.9), thus proving Lemma 6.2. ∎

To complete the proof of Theorem 6.1 we need to show that if $\lambda$ is a
geodesic measured lamination such that
$\tau_{\varepsilon_{n},W}\underset{n\to\infty}{\longrightarrow}\llbracket\lambda\rrbracket$
for some sequence $\varepsilon_{n}\to 0$ then $\lambda=\lambda_{s,t}$ with
$(s,t)\in[m_{p},M_{p}]\times[m_{q},M_{q}]$. Recall that just like in [HŠ21] we
have that $\mathbb{M}_{I,J}(\varepsilon)\to 2/3$ as $\varepsilon\to 0$
whenever $\Sigma\cap(I\times J)=\gamma$ for some
$\gamma\in\Sigma\setminus{(g_{1}\cup g_{2})}$. Hence it follows from Corollary
2.8, Lemma 2.9 and Lemma 6.2 that to prove Theorem 6.1 it is enough to show
that the following inequalities hold:

$\displaystyle\limsup_{\varepsilon\to
0}\mathbb{M}_{I_{0},J_{i}}(\varepsilon)\leq\begin{cases}M_{p},&\mbox{ if
}i=0,\\\ M_{q},&\mbox{ if }i=1,\end{cases}$
$\displaystyle\liminf_{\varepsilon\to
0}\mathbb{M}_{I_{0},J_{i}}(\varepsilon)\geq\begin{cases}m_{p},&\mbox{ if
}i=0,\\\ m_{q},&\mbox{ if }i=1.\end{cases}$

However, these follow from inequalities (6.10) and (6.14) applied to the
Example 3.5. ∎

###  6.2. Limit sets of dimension $k\in[2,\infty]$

Let $2\leq k<\infty$. For all $p,q\in(1,\infty)$ let
$W_{p,q}\subset\\{0<\Re(z)<3\\}$ denote the domain constructed in Section 6.1
above. Given a domain $W\subset\mathbb{C}$ and $z\in\mathbb{C}$ we denote
$W+z=\\{w+z|w\in W\\}$, i.e. the Minkowski sum of $W$ and $z$, or equivalently
$z$ translate of $W$.

Suppose $p_{j}\in(1,\infty)$ for $j\in\\{1,\ldots,k\\}$ and $a\in(0,1)$. For
$n\geq 1$, let

$\displaystyle a_{j,n}=a^{p_{j}^{2n}},\quad
b_{j,n}=a_{j,n}^{1/p}=a^{p_{j}^{2n-1}},\quad c_{j,n}=3-a^{p_{j}^{2n}},\quad
d_{j,n}=3-a^{p_{j}^{2n-1}}.$

Also, let $C_{j,n}=(a_{j,n},b_{j,n})\times(-\infty,\infty)$ and
$C^{\prime}_{j,n}=(d_{j,n},c_{j,n})\times(-\infty,\infty)$. We define the
domains $W_{j}:=W_{p_{j},p_{j}}+3j$, where

(6.17) $\displaystyle W_{p_{j},p_{j}}=\bigcup_{n=0}^{\infty}(C_{j,n}\cup
C_{j,n}^{\prime})\cap\\{z\,|\,0<\Re(z)<3,\Im(z)<0\\}$

Figure 6.1. Construction of domain $\mathcal{W}_{k}$.

Note that the domains $W_{j}$ are pairwise disjoint, but
$\overline{W_{j}}\cap\overline{W_{j+1}}=\\{z\in\mathbb{C}\,|\,\Re z=3j\\}$.
Define the domain $\mathcal{W}_{k}$ as follows:

(6.18)
$\displaystyle\mathcal{W}_{k}=\bigcup_{j=1}^{k}W_{j}\cup((0,3k)\times(-1,0)).$

Thus $\mathcal{W}_{k}$ is obtained from $\cup_{j=1}^{k}W_{j}$ by “adding” the
vertical intervals $\\{3j\\}\times(-1,0)$ which can be thought of as
“channels” connecting $W_{j}$ and $W_{j+1}$ in $\mathcal{W}_{k}$.

Let $\psi_{k}:\mathbb{D}\to\mathcal{W}_{k}$ be a conformal map such that

(6.19) $\displaystyle\begin{split}\lim_{z\to 0}\psi_{k}^{-1}(z)&=-1,\\\
\lim_{z\to 3k}\psi_{k}^{-1}(z)&=1,\\\
\lim_{\Im(z)\to-\infty}\psi_{k}^{-1}(z)&=-i.\end{split}$

We define the geodesics $\gamma_{n,j}^{\pm}$ and $\tilde{\gamma}_{n,j}^{\pm}$
for $j\in\\{1,\ldots,k\\}$ and $n\geq 0$ analogously to Section 6.1, i.e., for
every chimney in $\mathcal{W}_{k}\cap W_{j}$ we have two corresponding
geodesics in the unit disk with a common endpoint. We let

(6.20)
$\displaystyle\lambda_{\mathcal{W}_{k}}=\sum_{j=1}^{k}\left(\sum_{n=0}^{\infty}(\delta_{\gamma_{n,j}^{+}}+\delta_{\gamma_{n,j}^{-}})+\sum_{n=0}^{\infty}(\delta_{\tilde{\gamma}_{n,j}^{+}}+\delta_{\tilde{\gamma}_{n,j}^{-}})\right).$

Letting $\xi_{j}:=\lim_{w\to 3j}\psi_{k}^{-1}(w)$ for $j\in\\{0,\ldots,k\\}$,
and

$\displaystyle\zeta_{j}$
$\displaystyle=\lim_{\begin{subarray}{c}\Im(z)\to\infty\\\ z\in
W_{j}\end{subarray}}\psi_{k}^{-1}(w),\quad$

for $j\in\\{1,\ldots,k\\}$ we also denote by $g_{j}$ and $g_{j}^{\prime}$ the
hyperbolic geodesics in $\mathbb{D}$ connecting $\zeta_{j}$ to $\xi_{j-1}$ and
$\xi_{j}$, respectively.

Recall that real numbers $\theta_{1},\ldots,\theta_{k}$ are _rationally
independent_ if the equation $r_{1}\theta_{1}+\ldots+r_{k}\theta_{k}=0$ with
integer coefficients $r_{j}$ holds only if $r_{1}=\ldots=r_{k}=0$.

######  Theorem 6.3.

Suppose the collection of numbers $\log p_{1},\ldots,\log p_{k}$ is rationally
independent. If the domain $\mathcal{W}_{k}$ is defined as in (6.18) then the
generalized Teichmüller geodesic
$\varepsilon\mapsto\tau_{\varepsilon,\mathcal{W}_{k}}$ diverges and the
accumulation set $\Lambda$ in $\partial_{\infty}T(\mathbb{D})$ can be
described as follows:

(6.21)
$\displaystyle\Lambda=\left\\{\llbracket\lambda_{s_{1},\ldots,s_{k}}\rrbracket:(s_{1},\ldots,s_{k})\in\prod_{j=1}^{k}[m_{p_{j}},M_{p_{j}}]\right\\},$

where

(6.22)
$\displaystyle\lambda_{s_{1},\ldots,s_{k}}=\sum_{j=1}^{k}s_{j}(\delta_{g_{j}}+\delta_{g_{j}^{\prime}})+\frac{2}{3}\lambda_{\mathcal{W}_{k}}.$

The proof of Theorem 6.3 is a generalization of the proof of Theorem 6.1. The
key difference is that we use the classical Kronecker approximation theorem
about the density of the sequence
$(\\{\theta_{1}n\\},\ldots,\\{\theta_{k}n\\})$ in the $k$-dimensional unit
cube when $\theta_{1},\ldots,\theta_{k}$ are rationally independent, instead
of the the fact that every orbit $\\{T^{\circ n}_{\theta}(x)\\}$ of an
irrational rotation is dense in $\mathbb{S}^{1}$. For this reason we provide a
detailed sketch of the proof Theorem 6.3, and leave the verification of the
few missing details to the reader.

###### Sketch of the proof of Theorem 6.3.

The inclusion $\subset$ in (6.21) follows from the inequalities
$m_{j}\leq\liminf_{\varepsilon\to 0}M_{I_{j},J_{j}}(\varepsilon)$ and
$\liminf_{\varepsilon\to 0}M_{I_{j},J_{j}}(\varepsilon)\leq M_{j}$ just like
in Theorem 6.1.

Let $\Sigma_{k}$ denote the support of a measured lamination
$\lambda_{s_{1},\ldots,s_{k}}$, which is independent of the choice of the
weights $s_{j}$. The proof of the inclusion $\supset$ in (6.21), just like for
Theorem 6.1, comes down to showing that for every choice of
$(s_{1},\ldots,s_{k})\in\prod_{j=1}^{k}[m_{p_{j}},M_{p_{j}}]$ there exists a
sequence $\varepsilon_{n_{k}}$ so that if the only geodesics from $\Sigma_{k}$
contained in the box $I\times J$ are either $g_{j}$ or $g_{j}^{\prime}$ for
some $j\in\\{1,\ldots,k\\}$ then

(6.23)
$\displaystyle\lim_{n\to\infty}\mathbb{M}_{I,J}(\varepsilon_{n_{k}})=s_{j}.$

In the cases when $\Sigma_{k}\cap(I\times
J)=g\notin\\{g_{j},g_{j}^{\prime}\\}$ and $\Sigma_{k}\cap(I\times
J)=\emptyset$. In these cases we have that
$\lim_{n\to\infty}\mathbb{M}_{I,J}(\varepsilon_{n})$ is equal to either $2/3$
or $0$, respectively, which is proved just like in Theorem 6.1 or rather in
[HŠ21].

Let $I_{j}=\partial\mathcal{W}_{k}\cap\\{3j<\Re z<3j+1\\}\cap\\{\Im z>0\\}$,
for $j\in\\{0,\ldots,k-1\\}$, $J_{j}=\\{3j\\}\times(-\infty,-1),$ for
$j\in\\{0,\ldots,k\\}$. Then, like in the proof of Theorem 6.1 it is enough to
show that for $j\in\\{0,\ldots,k-1\\}$ we have

(6.24)
$\displaystyle\lim_{n\to\infty}\mathbb{M}_{I_{j},J_{j}}(\varepsilon_{n})=\lim_{n\to\infty}\mathbb{M}_{I_{j},J_{j+1}}(\varepsilon_{n})=s_{j}.$

As in the proof of Theorem 6.1 we choose $\alpha_{1}$ so that
$\Phi_{p_{1}}(\alpha_{1})=s_{1}$, let $\varepsilon_{n}=b_{1,n}^{\alpha_{1}}$,
and observe that for every $j\in\\{1,\ldots,k\\}$ and $n\geq 1$ there is a
unique $m_{j}=m_{j,n}\in\mathbb{N}$ such that
$\varepsilon_{n}=b_{1,n}^{\alpha_{1}}=a^{\alpha_{1}p_{j}^{2n-1}}\in(a^{p_{j}^{2m_{j}+1}},a^{p_{j}^{2m_{j}-1}}].$
Therefore, $\exists\,\,\beta_{j,n}\in[0,1)$ such that
$\varepsilon_{n}=a^{q^{2m_{j}-1+2\beta_{j,n}}}$, or
$p_{1}^{2n-1+\log_{p_{1}}\alpha_{1}}=q^{2m_{j}-1+2\beta_{j,n}}.$ Solving for
$\beta_{j,n}$ we obtain

$\displaystyle\beta_{j,n}$ $\displaystyle=\left(2n-1+\frac{\log\alpha}{\log
p_{1}}\right)\frac{\log p_{1}}{2\log
p_{j}}+\frac{1}{2}-m_{j}=\theta_{j}n+\sigma_{j}-m_{j},$

where $\theta_{j}:=\log_{p_{j}}p_{1}$ and
$\sigma_{j}:=\frac{1}{2}(1+\log_{p_{j}}\frac{\alpha_{1}}{p_{1}})$. Hence, for
$j\in\\{1,\ldots,k\\}$ we have

(6.25) $\displaystyle\beta_{j,n}\equiv\theta_{j}n+\sigma_{j}(\mathrm{mod}1).$

Next we use the classical theorem of Kronecker, see [HW79, Thm. 443, p. 382].

######  Theorem 6.4 (Kronecker’s density theorem).

If $\theta_{1},\theta_{2},\ldots,\theta_{k}$ are rationally independent,
$x_{1},\ldots,x_{k}$ are arbitrary, and $N$ and $\varepsilon$ are positive,
then there are integers $n>N$, and $p_{1},\ldots,p_{k}$ such that

(6.26) $\displaystyle|n\theta_{j}-p_{j}-x_{j}|<\varepsilon,$

for all $j\in\\{1,\ldots,k\\}$.

Therefore, for every choice of $\beta_{1},\ldots,\beta_{k}$ there are
sequences of natural numbers $n_{k}$ and $p_{j,k}$ such that

$|n_{k}\theta_{j}-p_{j,k}-\sigma_{j}-\beta_{j}|<1/k$

for all $j\in\\{1,\ldots,k\\}$. Hence, for this choice of $n_{k}$ we have that
$\beta_{j,n_{k}}\to\beta_{j}$ as $k\to\infty$ for all $j\in\\{1,\ldots,k\\}$.
Moreover, arguing as in the proof of Lemma 6.2 (the part after equation
(6.14)), we can see that for an arbitrrary choice of $s_{j}$ we can choose the
sequence $n_{k}$ so that for every $j\in\\{1,\ldots,k\\}$ we have

$\displaystyle\lim_{k\to\infty}\mathbb{M}_{I_{j},J_{j}}(\varepsilon_{n_{k}})=\lim_{k\to\infty}\Phi_{p_{j}}(p_{j}^{2\beta_{j,n_{k}}})=\Phi_{p_{j}}(p_{j}^{2\beta_{j}})=s_{j}.$

This proves the inclusion $\supset$ in (6.21) and completes the proof of the
theorem. ∎

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