# Separation of horocycle orbits on moduli space in genus 2

John Rached

###### Abstract

We prove a quantitative closing lemma for the horocycle flow induced by the
$\mathrm{SL}(2,\mathbb{R})$-action on the moduli space of Abelian
differentials with a double-order zero on surfaces of genus 2. The proof
proceeds via construction of a Margulis function measuring the discretized
fractal dimension of separation of a horocycle orbit of a point from itself,
in a direction transverse to the $\mathrm{SL}(2,\mathbb{R})$-orbit. From this,
we deduce that small transversal separation guarantees the existence of a
nearby point with a pseudo-Anosov in its Veech group. This is reminiscent of
the initial dimension phases in Bourgain-Gamburd for random walks on compact
groups, Bourgain-Lindenstrauss-Furman-Mozes for quantitative equidistribution
in tori, and quantitative equidistribution of horocycle flow for a product of
$\mathrm{SL}(2,\mathbb{R})$ with itself due to Lindenstrauss-Mohammadi-Wang,
and multiple other works.

##  1 Introduction

Dynamical closing lemmas have been investigated for several decades, in a
broad variety of contexts. In general terms, "closing lemma" refers to
perturbing an initial point in a dynamical system, whose orbit is recurrent to
a compact set, to find a point of small distance from the initial point that
lies on a periodic orbit. "Quantitative closing lemma" generally indicates a
bound in terms of the time parameter on the period of the closed orbit
achieved. Two results that have inspired substantial research activity are
Pugh’s closing lemma for $C^{1}$-diffeomorphisms of compact smooth manifolds
[35], and Anosov’s closing lemma [25] for hyperbolic sets of a flow. An open
line of research in the direction of the former is to achieve such a closing
lemma for higher orders of smoothness than $C^{1}$ [1]. This was resolved for
the class of Hamiltonian diffeomorphisms of closed surfaces by Asaoka and Irie
[2]. A quantitative result in this spirit, for Reeb orbits on contact
3-manifolds, was recently shown by Hutchings [23].

In Teichmüller dynamics, the diagonal flow induced by the
$\mathrm{SL}(2,\mathbb{R})$-action on the Hodge bundle over moduli space
projects to geodesics for the Teichmüller metric on moduli space. This action
is uniformly hyperbolic (Anosov) on compact sets; Hamendstädt [20] and Eskin-
Mirzakhani-Rafi [14] have independently proven closing lemmas for this flow. A
version of this lemma for affine invariant submanifolds of strata in the Hodge
bundle is used by Wright in [46], to prove that such manifolds are defined
over number fields. A similar version of this lemma is invoked in [15] to show
that the Zariski closure of the monodromy of the Kontsevich-Zorich cocycle
restricted to the tangent bundle of an affine invariant manifold is the full
group of endomorphisms preserving the symplectic form on the manifold.

Our concern in this article is with the action of a different one-parameter
subgroup of $\mathrm{SL}(2,\mathbb{R})$ acting on a stratum of the Hodge
bundle, namely, the action of the unipotent $U=\left\\{u_{s}\ |\
s\in\mathbb{R}\right\\}$, where $u_{s}=\begin{pmatrix}1&s\\\
0&1\end{pmatrix}$. The $u_{s}$-action remains poorly understood, in contrast
to the full $\mathrm{SL}(2,\mathbb{R})$-action, where a complete
classification of invariant and stationary measures was obtained by Eskin-
Mirzkakhani [11]. Full $\mathrm{SL}(2,\mathbb{R})$-orbit closures are known to
be affine invariant submanifolds of a stratum of the Hodge bundle [12], and
are in fact algebraic varieties [16]. On the other hand, $u_{s}$-orbit
closures can have fractional Hausdorff dimensions [8] and exhibit other
features distinctive from the homogeneous theory [7]. Nevertheless, we show
that in the single double-zero setting in genus 2, small transversal
separation of a geodesic push of a horocycle orbit guarantees proximity to a
periodic orbit, establishing a resonance in moduli space for the "initial
dimension phase" in Bourgain-Furman-Lindenstrauss-Mozes [6], Lindenstrauss-
Margulis [27], Yang [47] and Lindenstrauss-
Mohammadi-Wang [28]. The dichotomy between large dimension of transversal
separation and nearly lying on a periodic orbit was used in the aforementioned
works to establish quantitative equidistribution results in those settings;
this is our original motivation as well.

###  1.1 Overview

For $G$ a connected Lie group, and $\Gamma$ a lattice, Ratner’s seminal
theorems [38], [37], [39] provide a classification for orbits of all points in
$G/\Gamma$ under the action of a subgroup of $G$ generated by unipotent
elements. Ratner’s theorems rely on the pointwise ergodic theorem, making
these results difficult to effectivize in most cases. Green and Tao prove
equidistribution for nilflows with polynomial error rates [19]. When the
unipotent subgroup is horospherical, as in the case of the horocycle flow on
$\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})$, effective
equidistribution for long unipotent orbits is known, and in the case of
$\mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z})$ can be deduced from
effective ergodicity of the horocycle flow as discussed in an original work of
Ratner [36]. Recently, there has been activity in search of effective accounts
of equidistribution of both flows and closed unipotent orbits beyond the
horospherical and nilflow settings. One starting point of this was with the
work of Einsiedler, Margulis and Venkatesh who achieve effective
equidistribution of closed unipotent orbits with polynomial error rate in a
general semisimple homogeneous (arithmetic) setting [9]. The case of effective
equidistribution with polynomial error rates for unipotent orbits on
arithmetic quotients
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ was also done
recently by Lindenstrauss, Mohammadi and Wang [29]. Using different methods,
Str$\"{o}$mbergsson obtained polynomial effective equidistribution for a
nonhorospherical action on the quotient of
$\mathrm{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2}$ by
$\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$ [44].

The results above leverage arithmeticity of the lattice $\Gamma$ in a crucial
way. The overarching idea is to establish analogs of Liouville’s theorem,
which states roughly that irrational algebraic numbers are badly approximable
by rational numbers. We provide some details to illustrate this concept at
work in the setting of Bourgain-Furman-Lindenstrauss-Mozes. Let $\Gamma$ be a
semigroup of $d\times d$ non-singular integer matrices. If the natural action
on the torus $\mathbb{T}^{d}$ is strongly irreducible, then the action of
$\Gamma$ is ergodic with respect to Haar measure. In the case $d=1$ and
$\Gamma$ is Abelian, Furstenberg [18] showed that provided $\Gamma$ contains
no finite index cyclic groups, $\Gamma\cdot x$ is dense for all
$x\in\mathbb{R}\setminus\mathbb{Q}$. Bourgain, Furman, Lindenstrauss and Mozes
[6] considered the situation, where, along with other technical assumptions,
the natural action of $\Gamma$ on $\mathbb{R}^{d}$ is assumed to be strongly
irreducible (this is strictly stronger than the assumption that $\Gamma$
acting on $\mathbb{T}^{d}$ is strongly irreducible). When $\Gamma$ acts
strongly irreducibly on $\mathbb{R}^{d}$, and $\nu$ is a probability measure
on $\Gamma$, the top Lyapunov exponent
$\lambda_{1}(\nu)=\lim\limits_{n\mapsto\infty}\frac{1}{n}\mathrm{log}\
\lvert\lvert g_{1}g_{2}\cdot\cdot\cdot g_{n}\rvert\rvert$ is $\nu$-almost
surely positive. Given a probability measure $\nu$ on $\Gamma$ and a
probability measure $\mu$ on $\mathbb{T}^{d}$, one can define a probability
measure on $\mathbb{T}^{d}$ by
$\nu\ast\mu=\sum\limits_{g\in\Gamma}\nu(g)g_{*}\mu$. Under some additional
mild assumptions, including a moment condition on $\nu$, Bourgain-Furman-
Lindenstrauss-Mozes [6] prove that if $\nu$ is a fixed probability measure
supported on $\Gamma<\mathrm{SL}_{d}(\mathbb{R})$, then for any
$0<\lambda<\lambda_{1}$, there is a constant $C=C(\nu,\lambda)$, so that if
for a point $x\in\mathbb{T}^{d}$ the measure $\mu_{n}=\nu^{(n)}\ast\delta_{x}$
satisfies that for some $a\in\mathbb{Z}^{d}\setminus\left\\{0\right\\}$

$\bigl{\lvert}\hat{\mu}_{n}(a)>t>0\bigr{\rvert}\ \ \textrm{with}\ \
n>C\cdot\mathrm{log}\frac{2\lvert\lvert a\rvert\rvert}{t},$

then $x$ admits a rational approximation $\frac{p}{q}$ with
$p\in\mathbb{Z}^{d}$ and $q\in\mathbb{Z}^{+}$ satisfying

$\Bigl{\lvert}\Bigl{\lvert}x-\frac{p}{q}\Bigr{\rvert}\Bigr{\rvert}<e^{-\lambda
n}\ \ \mathrm{and}\ \lvert q\rvert<\left(\frac{2\lvert\lvert
a\rvert\rvert}{t}\right)^{C}.$

A key step in the proof of this theorem is to start with an absolute lower
bound on a single Fourier coefficient of the measure
$\mu_{n}=\nu^{\ast(n)}\ast\delta_{x}$ and further conclude for a chosen $m<n$,
$\mu_{n-m}$ has a "rich" set of Fourier coefficients larger than a polynomial
in $t$. Starting with some
$a_{0}\in\mathbb{Z}^{d}\big{\backslash}\left\\{0\right\\}$ with
$\lvert\hat{\mu}_{n}(a_{0})>t_{0}$ for large enough $n$, the idea is to prove
that for $t=t_{0}^{p}$, the set

$A_{n-m_{1},t}=\left\\{a\in\mathbb{Z}^{d}:\
\bigl{\lvert}\hat{\mu}_{n-m_{1}}(a)\bigr{\rvert}>t\right\\}$

is a thick set in $\mathbb{Z}^{d}$. For a subset $E\subset\mathbb{Z}^{d}$, let
$\mathcal{N}(E;R)$ denote the covering number of $E$ by balls of radius $R$.
The following is the crucial step.

######  Lemma 1.1 (B-F-L-M [6]).

There exists a large $N$ and an exponentially smaller $R$, so that the number
of $R$-balls needed to cover the intersection of a rich set of Fourier
coefficients with a large box is greater than a polynomial. Namely,
$\mathcal{N}\left(A_{m-n_{1},t_{0}^{p}}\cap[-N,N]^{d};R\right)>t_{0}^{p}\left(N/R\right)^{d}$.

By an analog of Wiener’s lemma, Lemma 1.1, gives a "small-dimensional"
separation of points on the torus. This information is bootstrapped to reach a
dimension of separation equal to the dimension of the torus.

Returning to the Lie group setting, this general scheme of achieving a small
initial dimension of separation and boostrapping the dimension is successfully
employed by Lindenstrauss-Mohammadi and Lindenstrauss-Mohammadi-Wang to
achieve polynomially effective density and polynomially effective
equidistribution of one-parameter unipotent flows in arithmetic quotients of
$\mathrm{SL}(2,\mathbb{C})$ and
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$, and polynomial
rates for the Oppenheim conjecture [28], [26], [30]. In these works, the
authors prove quantitative closing lemmas (for instance, Proposition 4.8 in
[28]) which are themselves similar to Lemma 5.2 in Lindenstrauss-Margulis
[27]. In [28], [26], [30], these closing lemmas are used to produce a small
transversal dimension of separation in the unipotent direction, for unipotent
orbits not too close to a periodic orbit. This serves as the analogue of Lemma
1.1, and this dimension is bootstrapped to achieve dimension close to that of
a horospherical subgroup, for which there exist methods to deduce effective
density and effective equidistribution. We proceed to state our main theorem,
which is a direct analog of Proposition 4.8 in [28], a direct analog of
Proposition 13.1 in [9] and similar in spirit to Lemma 1.1.

Let $S$ = $S_{g}$ be a surface of genus $g$, and let $\mathcal{M}_{g}$ be the
moduli space of Riemann surfaces homeomorphic to $S$. Let
$\Omega\mathcal{M}_{g}$ be the space of Abelian differentials (holomorphic
1-forms $\omega$). Let $\Omega_{1}\mathcal{M}_{g}\subset\Omega\mathcal{M}_{g}$
be the unit-area locus of $\Omega\mathcal{M}_{g}$. There is an
$\mathrm{SL}(2,\mathbb{R})$-action on $\Omega_{1}\mathcal{M}_{g}$ given by the
action of $\mathrm{SL}(2,\mathbb{R})$ on $\mathbb{R}^{2}$ corresponding to
"multiplication by $A\in\mathrm{SL}(2,\mathbb{R})$" on the atlas of charts to
$\mathbb{R}^{2}$ determined by the quadratic differential. By Teichmüller
theory, the orbits of the diagonal flow $a_{t}=\begin{bmatrix}e^{t}&0\\\
0&e^{-t}\end{bmatrix}$ project to Teichmüller geodesics under the natural
projection map $\pi:\Omega_{1}\mathcal{M}_{g}\rightarrow\mathcal{M}_{g}$.
$\Omega\mathcal{M}_{g}$ has a natural stratification: we say
$(X,\omega)\in\Omega\mathcal{M}_{g}$ is of type
$\sigma=\left(p_{1},...,p_{k}\right)$ if $\omega$ has zeroes of order
$\left\\{p_{i}\right\\}$. The space of all Abelian differentials of type
$\sigma=\left(p_{1},...,p_{k}\right)$ in $\Omega\mathcal{M}_{g}$ will be
denoted by $\Omega\mathcal{M}_{g}\left(\sigma\right)$. This is a complex-
analytic orbifold of real dimension $4g+2k-1$. It also has the structure of a
complex algebraic variety. Let $\mathcal{T}_{g}$ be the Teichmüller space, the
orbifold universal cover of $\mathcal{M}_{g}$. Let $\Omega\mathcal{T}_{g}$ and
$\Omega_{1}\mathcal{T}_{g}$ be defined similarly as above. We have an orbifold
covering map:

$\pi:\Omega_{1}\mathcal{T}_{g}\left(\sigma\right)\rightarrow\Omega_{1}\mathcal{M}_{g}\left(\sigma\right).$

Moreover, if $V\subset\Omega_{1}\mathcal{M}_{g}\left(\sigma\right)$ is a
properly immersed manifold, the pre-image $\pi^{-1}(V)$ is a properly immersed
manifold of the same dimension. The lowest dimensional
$\mathrm{SL}(2,\mathbb{R})$-invariant manifolds that can be cut out locally by
real-linear equations are of particular relevance to us. Specifically, we are
interested in the orbit closures of
$\pi\left(\mathrm{SL}(2,\mathbb{R})\cdot\omega\right)$ for special choices of
$(X,\omega)\in\Omega_{1}\mathcal{T}_{g}\left(\sigma\right)$. For
$\mathrm{Mod}(S)$ the mapping class group of $S=S_{g}$, it follows from a
classical result of of Smillie-Weiss [42] that if
$\Gamma_{\omega}\leq\mathrm{Mod}(S)$ is the set-wise stabilizer of the lift of
$\pi\left(\mathrm{SL}(2,\mathbb{R})\cdot\omega\right)$ to the connected
component
$\widetilde{{\pi^{-1}\circ{\pi\left(\mathrm{SL}(2,\mathbb{R})\cdot\omega\right)}}}$
of $\Omega_{1}\mathcal{T}_{g}\left(\sigma\right)$ containing $(X,\omega)$,
then

$\widetilde{{\pi^{-1}\circ{\pi\left(\mathrm{SL}(2,\mathbb{R})\cdot\omega\right)}}}=\Gamma_{w}\cdot\left(\overline{\mathrm{SL}(2,\mathbb{R})\cdot\omega}\right)$

when $(X,\omega)$ has lattice-stabilizer. In this case, $(X,\omega)$ is
referred to as a Veech surface. Veech surfaces have special dynamical
properties. In particular, for such $\omega$,
$\pi\left(\mathrm{SL}(2,\mathbb{R})\cdot\omega\right)$ is a closed orbit.
Furthermore, the image of $\mathrm{SL}(2,\mathbb{R})\cdot\omega$ under

$\Omega_{1}\mathcal{T}_{g}\left(\sigma\right)\xrightarrow{\pi}\Omega_{1}\mathcal{M}_{g}\left(\sigma\right)\rightarrow\mathcal{M}_{g}$

is an algebraic curve $V$ in $\mathcal{M}_{g}(S)$, called a Teichmüller curve.
Let $e^{-0.01t}<\beta=e^{-\kappa t}<1$ for a fixed $\kappa=1/D>0$ where $D$
will be specified below. For every $t\geq 0$, define

$E_{t}=B_{\beta}\cdot
a_{t}\cdot\left\\{u_{r}:r\in[0,1]\right\\}\subset\mathrm{SL}(2,\mathbb{R})$

where
$B_{\beta}:=\left\\{u_{s}^{\intercal}:|s|\leq\beta\right\\}\cdot\left\\{a_{t}:|t|\leq\beta\right\\}$
and $u_{s}^{\intercal}$ is the transpose of $u_{s}$. In words, this is a
"smearing" by a small amount of $\mathrm{SL}(2,\mathbb{R})$ of a geodesic push
of length $t$ of a length-1 horocycle segment. The principal reason for
stating our separation theorem with a smearing approximation of geodesic
translates of unit-length horocycle orbits is for potential application to
quantitative equidistribution theorems. For a concise discussion of this, see
Section 6 of [29].

The following Theorem shows that, up to removing an exceptional set of very
small measure, geodesic pushes of length-1 horocycle segments are separated in
a direction tranverse to the $\mathrm{SL}(2,\mathbb{R})$-orbit of a point (up
to "smearing"), so long as the orbit of this point doesn’t come too close to a
Teichmüller curve, with a quantitative relationship between the proximity to
the Teichmüller curve and the size of a hyperbolic element the curve contains
in its Veech group. We state our Main Theorem.

######  Theorem 1.2.

Let $\Omega_{1}\mathcal{M}_{2}(2)$ be the stratum of unit-area Abelian
differentials of genus 2 with a single zero. There exist $D_{0}\geq 0,\alpha$
depending only on the stratum, satisfying the following. Suppose $D\geq
D_{0}+1$. For any $x=(X,\omega)\in\Omega_{1}\mathcal{M}_{2}(2)$, there exists
$t^{\prime}$ depending only on the injectivity radius of $x$ such that for all
$t\geq t^{\prime}$, there exists a function $f_{t}:E_{t}\cdot
x\rightarrow(0,\infty)$ so that at least one of the following is true.

  1. 1.

There is a subset $I\subset[0,1]$ with
$\bigl{\lvert}[0,1]\big{\backslash}I\bigr{\rvert}\ll\beta^{\alpha}$ such that
for all $s\in I$ we have the following

    1. (a)

$\mathrm{inj}(a_{8t}u_{s}x)\geq\beta$,

    2. (b)

$h\rightarrow h\cdot a_{8t}u_{s}x$ is an injective map
$E_{t}\mapsto\Omega_{1}\mathcal{M}_{2}(2)$,

    3. (c)

for all $z\in E_{t}\cdot a_{8t}u_{s}x$, we have

$f_{t}(z)\leq e^{Dt}.$

  2. 2.

There is
$x^{\prime}=(X^{\prime},\omega^{\prime})\in\Omega_{1}\mathcal{M}_{2}(2)$ such
that $x^{\prime}$ lies on a Teichmüller curve $V$ and
$d_{\mathcal{T}}\left(z^{\prime},x^{\prime}\right)\leq e^{(-D+D_{0})t}$ for
some $z^{\prime}\in a_{8t}u_{s^{\prime}}\cdot x$, where
$d_{\mathcal{T}}(\cdot,\cdot)$ is the Teichmüller distance and
$s^{\prime}\in[0,1]$. The Veech group of $x^{\prime}$ contains a hyperbolic
element $\gamma$ with $\mathrm{log}(l(\gamma))\leq e^{Dt}$, where $l(\gamma)$
is the translation length.

We expect Theorem 1.2 to fail for all other strata
$\Omega_{1}\mathcal{M}_{g}(\sigma)$, though we supply no proof in this
article. Essentially, this is due to the existence of non-lattice Veech groups
in other strata which are infinitely generated, constructed by Hubert and
Schmidt [22], and McMullen [33]. However, we make the following conjecture,
which we believe would serve as a substitute for the "initial dimension phase"
for all strata of moduli space, opening the door to adopting the schema
described above to establish quantitative equidistribution results for
horocycle flow in all strata. In particular, combining Theorem 1.2 with
results of Sanchez on effective equidistribution of large dimensional measures
[40] would leave only the ’bootstrapping’ step of [28]. Since [40] applies to
all strata, a similar obstacle would remain were the following conjecture to
be established.

######  Question 1.3.

Does there exist a sequence of Veech groups
$\Gamma_{k}\in\Omega_{1}\mathcal{M}_{g}(\sigma)$ in any stratum with critical
exponents approaching one?

An affirmative answer to the analogous statement for Kleinian groups has
recently been established by [43], with powerful techniques that may be
applicable to Teichmüller dynamics.

The paper is organized as follows. In Section 2, we review fundamental facts
on the geometry of moduli space, which we rely on to adapt the proofs of
homogeneous closing lemmas to this context. In Section 3, we use results of
Minsky-Weiss to state quantitative-nondivergence that establishes enough
recurrence to compact sets to reduce the statement of Theorem 1.2 to a
dichotomy. In Section 4, we construct a Margulis function that measures the
transversal (to the $\mathrm{SL}(2,\mathbb{R})$ direction) separation of an
orbit from itself, and show the group of elements in
$\mathrm{SL}(2,\mathbb{R})$ bringing an orbit close back to itself in the
tranverse direction cannot be unipotent. We conclude there must be a
hyperbolic element of $\mathrm{SL}(2,\mathbb{R})$ stabilizing a nearby
element, and since we are working in $\Omega_{1}\mathcal{M}_{2}(2)$, this
element must lie on a Teichmüller curve.

## Acknowledgements

The author thanks Asaf Katz for helpful discussions on homogeneous dynamics,
and Alex Eskin, Paul Apisa, Jon Chaika, Alex Wright and Jenya Sapir for
insightful comments.

##  2 Preliminaries

###  2.1 Constants and matrix norms

We will use the notation $B\ll C$ to mean $B\leq kC$, where $k$ is only
allowed to depend on the stratum. For a matrix
$M\in\mathrm{SL}(2,\mathbb{R})$, we will denote by $||\cdot||$ the Frobenius
norm. Recall the decomposition $M=KAK$ where $A$ is the diagonal group and
$K=SO(2)$ is the maximal compact rotation group. The $KAK$ decomposition
coincides with the singular value decomposition $M=U\Sigma V^{\perp}$ up to
changing signs of the diagonal entries of the matrices in the middle. We make
extensive use of the following elementary facts. For
$M_{1},M_{2}\in\mathrm{SL}(2,\mathbb{R})$, and using $\sigma_{min}(M)$ to
denote the smallest singular value of a matrix:

  1. 1.

$||M_{1}||=||M^{-1}_{1}||$

  2. 2.

$||M_{1}M_{2}||\leq||M_{1}||\cdot||M_{2}||$

  3. 3.

$||M_{1}M_{2}||\geq\sigma_{min}(M_{1})\cdot||M_{2}||$.

###  2.2 Teichmüller metric and Teichmüller curves

In this subsection, we briefly review the metric structures we will be
interested in, and the dynamics of Veech surfaces. All of the material here is
standard, and we will sometimes follow [31]. Let $S_{g}$ be a surface of genus
$g$, satisfying $3g-3>0$. Let $\phi_{1}:S_{g}\rightarrow S_{1}$,
$\phi_{2}:S_{g}\rightarrow S_{2}$, be two diffeomorphisms, where $S_{1}$ and
$S_{2}$ have finite area. Denote by $\sim$ the equivalence relation where
$\phi_{1}\sim\phi_{2}$ if $\phi_{1}\circ\phi^{-1}_{2}$ is isotopic to a
holomorphic diffeomorphism. Define

$\mathcal{T}_{g}=\big{\\{}\phi:S_{g}\rightarrow S\ |\ \text{S is finite area
},\phi\ \text{a diffeomorphism}\big{\\}}\big{/}\sim.$

Given $\psi\in\text{Diff}^{+}\left(S_{g}\right)$, the group of orientation-
preserving diffeomorphisms of $S_{g}$, we obtain a map
$\mathcal{T}_{g}\rightarrow\mathcal{T}_{g}$ by precomposition by $\psi$. By
the definition of the equivalence relation $\sim$, if $\psi$ is contained in
the normal subgroup $\text{Diff}^{0}\left(S_{g}\right)$ of diffeomorphisms
isotopic to a holomorphic diffeomorphism, this map defined by precomposition
is trivial. Therefore, we have an action of the mapping class group of $S_{g}$

$\text{Mod}\left(S_{g}\right):=\text{Diff}^{+}\left(S_{g}\right)\big{/}\text{Diff}^{0}\left(S_{g}\right)$

on $\mathcal{T}_{g}(S)$. The moduli space $\mathcal{M}_{g}$ of $S_{g}$ is
defined as

$\mathcal{M}_{g}:=\mathcal{T}_{g}\big{/}\text{Mod}\left(S_{g}\right).$

$\mathcal{T}_{g}$ is a real-analytic space, and is homeomorphic to
$\mathbb{R}^{6g-6}$. The cotangent space at $X\in\mathcal{T}_{g}$ can be
identified with $Q(X)$, the space of holomorphic quadratic differentials on
$X$. The dual space (tangent space at $X$) is naturally identified with the
space of harmonic Beltrami differentials on $X$, ${B}(X)$. Suppose
$\mu=\mu(z)d\bar{z}\otimes(dz)^{-1}\in B(X)$ and $q=q(z)dz^{2}\in Q(X)$. There
is a natural non-degenerate pairing on $\mathcal{T}_{g}$

$\langle{\mu,q\rangle}=\int_{X}\mu(z)q(z)|dz|^{2}.$

The norm
$||\mu||_{\mathcal{T}}=\mathrm{sup}\left\\{|\langle\\{\mu,q\rangle\\}:q\in
Q(X),||q||_{1}=\int_{X}|q|=1\right\\}$ gives the Teichmüller metric on
$\mathcal{T}_{g}$. This is the infinitesimal description of the Teichmüller
metric. The distance between two points in Teichmüller space has a simple
description in terms of dilatation. We have

$d_{\mathcal{T}}\left(\left(\phi_{1}:S_{g}\rightarrow
S_{1}\right),\left(\phi_{2}:S_{g}\rightarrow
S_{2}\right)\right)=\frac{1}{2}\underset{\phi}{\mathrm{inf}}\;\mathrm{log}\;K_{\phi}$

where $\phi:X_{1}\rightarrow X_{2}$ ranges over all quasi-conformal maps from
$X_{1}$ to $X_{2}$ isotopic to $\phi_{1}\circ\phi_{2}^{-1}$ and $K_{\phi}\geq
1$ is the dilatation coefficient. Since the definition of the metric only
depends on the holomorphic structure at $X$, the Teichmüller metric on
$\mathcal{T}_{g}$ descends to a metric on $\mathcal{M}_{g}$.

It is well-known that a pair $(X,q)$ with $q\in Q(X)$ and $q\neq 0$, generates
a holomorphic embedding from the hyperbolic plane $\mathbb{H}$ with Kobayashi
metric on $\mathbb{H}$ and Teichmüller metric on $\mathcal{T}_{g}$

$\tilde{f}:\mathbb{H}\rightarrow\mathcal{T}_{g}.$

$\tilde{f}$ is an isometry. Passing to
$\mathcal{M}_{g}=\mathcal{T}_{g}\big{/}\mathrm{Mod}(S_{g})$, we obtain a
complex geodesic

$f:\mathbb{H}\rightarrow\mathcal{M}_{g}.$

We summarize the constructions of $\tilde{f}$ and $f$, and the connection to
dynamics. For $t$ a complex parameter, the Riemann surface
$X_{t}=\tilde{f}(t)$ is characterized by the property that the complex
dilatation $\mu_{t}$ of the extremal quasi-conformal map
$\psi_{t}:X\rightarrow X_{t}$ is given by

$\mu_{t}=\left(\frac{i-t}{i+t}\right)\cdot\frac{\bar{q}}{|q|}.$ (2.1)

Now, assume $q$ is the square of a holomorphic $1$-form $w$, $q=\omega^{2}$.
Then, these complex geodesics have an interpretation in terms of the
$\mathrm{SL}(2,\mathbb{R})$ action on
$\Omega_{1}\mathcal{T}_{g}\left(\sigma\right)$. In particular, any holomorphic
$1$ -form $\omega$ yields, away from its zeroes, a flat metric Euclidean
metric $|\omega|$ and an atlas of charts $U_{i}\rightarrow\mathbb{C}$ whose
transition functions are translations. The $\mathrm{SL}(2,\mathbb{R})$-action
on $\Omega_{1}\mathcal{T}_{g}\left(\sigma\right)$ is defined by composition of
these charts with matrices $A\in\mathrm{SL}(2,\mathbb{R})$ acting linearly on
$\mathbb{C}\simeq\mathbb{R}^{2}$. $A\in\mathrm{SL}(2,\mathbb{R})$ can be
interpreted as an affine map in these coordinate charts
$A\cdot(X,\omega)=(X^{\prime},\omega^{\prime})$. This yields a quasi-conformal
map $f:X\rightarrow X^{\prime}$ with terminal holomorphic $1$-form
$\omega^{\prime}$. In the case that $A\in\mathrm{SO}(2,\mathbb{R})$, the map
$f$ is conformal, so the action descends to a faithful action of
$\mathrm{SL}(2,\mathbb{R})/\mathrm{SO}(2,\mathbb{R})$. One can show that this
descended action induces an isometric injection

$\mathrm{SL}(2,\mathbb{R})/\mathrm{SO}(2,\mathbb{R})(X,\omega)\rightarrow\mathcal{T}_{g}.$

Furthermore, composition with the projection
$\pi:\mathcal{T}_{g}\rightarrow\mathcal{M}_{g}$ defines the complex geodesic

$f:\mathbb{H}\rightarrow\mathcal{M}_{g}.$

One can check that if $t\in\mathbb{C}$ parametrizes the Riemann surfaces, the
complex dilatation of the extremal quasi-conformal map $\psi_{t}:X\rightarrow
X_{t}$ is given by

$\mu_{t}=\left(\frac{i-t}{i+t}\right)\cdot\frac{\bar{\omega}}{|\omega|}$ (2.2)

which accords with Equation 2.1. Furthermore, $f$ factors through the quotient
space $V=\mathbb{H}/\Gamma$, where
$\Gamma=\left\\{B\in\mathrm{Aut}(\mathbb{H}):f(B\cdot t)=f(t)\ \forall
t\right\\}$. In the case that $\Gamma$ is a lattice, we call the quotient map

$f:V\rightarrow\mathcal{M}_{g}$

a Teichmüller curve. $f$ is proper and generically injective. The image $f(V)$
is not a normal subvariety, but we will not distinguish between the
Teichmüller curve as a map $f$ and the normalization of $f(V)$. One can
interpret $\Gamma$ above directly in terms of flat geometry. Let
$\mathrm{Aff}^{+}(X,\omega)$ be the set of orientation-preserving affine
diffeomorphisms of $(X,\omega)$. For such an affine diffeomorphism
$\phi:(X,\omega)\mapsto(X,\omega)$, the derivative $D\phi$ is constant. Since
these diffeomorphisms are assumed to be orientation preserving, and also
preserve the area of $\omega$, the map $\phi\mapsto D\phi$ defines a map of
$\mathrm{Aff}^{+}(X,\omega)$ into $\mathrm{SL}(2,\mathbb{R})$. We will call
the image of this map the Veech group of $(X,\omega)$, and denote it by
$\mathrm{SL}(X,\omega)$. By Proposition 3.2 in [31], $\Gamma$ and
$\mathrm{SL}(X,\omega)$ are conjugate.

###  2.3 Period Coordinates and Holonomy Vectors

For any $(X,\omega)\in\Omega\mathcal{M}_{g}\left(\sigma\right)$, there exists
a triangulation $T$ of the underlying surface $X$ by saddle connections.
Denote the zeroes of $\omega$ by $Z(\omega)$, where $|Z(\omega)|=k$. One can
choose $h=2g+|Z(\omega)|-1$ directed edges $\left\\{w_{i}\right\\}_{i=1}^{h}$
of $T$, and an open neighborhood $U\subset\Omega\mathcal{M}_{g}(\sigma)$ of
$(X,\omega)$ in its stratum, so that there exists an open analytic embedding:

$\phi_{T,\omega}:U\rightarrow
H^{1}(X,Z(\omega),\mathbb{C})\cong\mathbb{C}^{2g+Z(\omega)-1}$

called the period map. It is defined by $(X,\omega)$ to be the relative class
$[\omega]\in H^{1}(X,Z(\omega),\mathbb{C})$ which satisfies
$\left<[\omega],w_{i}\right>=\int_{w_{i}}\omega$ for all $w_{i}$ viewed as
relative classes in $H_{1}(X,Z(\omega),\mathbb{C})$. Thus, we may think of
$[\omega]$ as a local coordinate on $\Omega\mathcal{M}_{g}(\sigma).$ For any
other geodesic triangulation $T^{\prime}$, the map
$\phi_{T^{\prime},\omega}\circ\phi^{-1}_{T,\omega}$ is linear. A consequence
of the uniformization theorem is that every element
$(X,\omega)\in\Omega\mathcal{M}_{g}\left(\sigma\right)$ can be presented in
the form

$(X,\omega)=(P,dz)\big{/}\sim$

for a polygon $P\subset\mathbb{C}$, and the $1$-form $dz$ on $\mathbb{C}$. The
reason for this is that one can construct a geodesic triangulation of the flat
surface $(X,|\omega|)$, with $Z(\omega)$ among its vertices. $X$ can then be
presented as a quotient $\sim$ of the edges of a collection of triangles. One
can check that the periods $\int_{w_{i}}\omega$ defining $\phi_{T,\omega}$
correspond to the vectors $\left\\{v_{i}\right\\}\in\mathbb{C}$, the edges of
the polygon $P$.

###  2.4 $\mathrm{SL}(2,\mathbb{R})$-action and Transversal Directions

For a pair $(X,\omega)$, one can define the area of $(X,\omega)$

$\mathrm{area}(X,\omega)=\frac{i}{2}\int_{X}\omega\wedge\overline{\omega}.$

We denote by
$\Omega_{1}\mathcal{M}_{g}(\sigma)\subset\Omega\mathcal{M}_{g}(\sigma)$ the
unit-area locus. Assume $\left\\{w_{i}\right\\}_{i=1}^{h}$ forms a symplectic
$\mathbb{Z}$-basis of $H_{1}(X,Z(\omega),\mathbb{Z})$. Choose a fundamental
domain $\mathcal{F}$ for the action of $\mathrm{Mod}\left(S_{g}\right)$ on
$\Omega_{1}\mathcal{T}_{g}(\sigma)$. The fundamental domain $\mathcal{F}$ is
then identified with the quotient space $\Omega_{1}\mathcal{M}_{g}(\sigma)$.
In period coordinates, there is a linear $\mathrm{SL}(2,\mathbb{R})$-action on
$\mathcal{F}$; in particular, for a $2\times h$ real-valued matrix $B$
representing $\phi_{T,\omega}$ at a point
$x=(X,\omega)\in\Omega\mathcal{M}_{g}(\sigma)$, and
$g\in\mathrm{SL}(2,\mathbb{R})$

$g\cdot x=gBA(g,x)$

where the right-hand side is just matrix multiplication and
$A:\mathrm{SL}(2,\mathbb{R})\times\Omega_{1}\mathcal{M}_{g}(\sigma)\rightarrow\mathrm{Sp}(2g,\mathbb{Z})\ltimes\mathbb{R}^{h}$
is the change of basis required to return $g\cdot x$ to the fundamental
domain. The map $A$ is called the Kontsevich-Zorich cocycle. The Kontsevich-
Zorich cocycle is not quite a cocycle in the dynamical sense due to the fact
that the automorphism group $\mathrm{Aut}(X,\omega)$ is non-trivial (i.e.,
$\Omega\mathcal{M}_{g}(\sigma)$ is an orbifold). However, by considering for
example, a level-3 structure [11], $\Omega_{1}\mathcal{M}_{g}(\sigma)$ has a
finite cover which is a manifold instead of an orbifold. Henceforth, we will
assume we are working in the finite cover and continue to use the notation
$\Omega_{1}\mathcal{M}_{g}(\sigma)$. In particular, for a pair
$(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$, the underlying Riemann
surface $X$ will have no non-trivial automorphisms. Since applying a rotation
matrix to $(X,\omega)$ doesn’t change the complex structure on $X$, we may
assume the Veech groups $\mathrm{SL}(X,\omega)$ contain no elliptic elements.
This will be relevant in Subsection 4.4.

For any subset $\mathcal{N}\subset\Omega_{1}\mathcal{M}_{g}(\sigma)$, define

$\mathbb{R}\mathcal{N}=\left\\{(X,t\omega)\ |\
(X,\omega)\in\mathcal{N},t\in\mathbb{R}\right\\}\subset\mathbb{R}\Omega_{1}\mathcal{M}_{g}(\sigma).$

Denote by $H^{1}$ the complex flat vector bundle over
$\mathbb{R}\Omega_{1}\mathcal{M}_{g}(\sigma)$ whose fiber over any
$(X,\omega)$ is $H^{1}(X,\mathbb{C})$, and by $H^{1}_{rel}$ the complex flat
vector bundle whose fiber over $(X,\omega)$ is
$H^{1}(X,Z(\omega),\mathbb{C})$. We will denote by $p:H^{1}_{rel}\mapsto
H^{1}$ the forgetful map from relative to absolute cohomology.

######  Definition 2.1.

We call a subset
$\tilde{\mathcal{N}}=\mathbb{R}\mathcal{N}\subset\mathbb{R}\Omega_{1}\mathcal{M}_{g}(\sigma)$
an affine invariant submanifold of
$\Omega_{1}\mathcal{M}_{g}(\sigma)$ if $\mathcal{N}$ is an analytic
submanifold of $\Omega_{1}\mathcal{M}_{g}(\sigma)$, and locally in period
coordinates, $\tilde{\mathcal{N}}$ is a complex-linear subspace of
$\mathbb{C}^{2g+|Z(\omega)|-1}$ given by linear equations with real
coefficients.

The following observation is helpful for our purposes.

######  Proposition 2.2.

The tangent bundle of $\tilde{\mathcal{N}}$ is determined in local period
coordinates by a subspace

$T_{\mathbb{C}}(\tilde{\mathcal{N}})=T_{\mathbb{R}}(\tilde{\mathcal{N}})\otimes_{\mathbb{R}}\mathbb{C}$

where $T_{\mathbb{R}}(\tilde{\mathcal{N}})\subset
H^{1}(X,Z(\omega),\mathbb{R})$. Further, the tangent space to the stratum at
$(X,\omega)\in\mathbb{R}\Omega_{1}\mathcal{M}_{g}(\sigma)$ is identified with
$H^{1}(X,Z(\omega),\mathbb{C})\cong
H^{1}(X,Z(\omega),\mathbb{R})\otimes_{\mathbb{R}}\mathbb{C}$ and the
$\mathrm{SL}(2,\mathbb{R})$-action is on $\mathbb{C}\cong\mathbb{R}^{2}$ under
this identification. In addition, if in local period coordinates $(X,\omega)$
is written as $x=u+iv$, for $u$ and $v$ column vectors, then the
$\mathbb{R}$-linear span of $u$ and $v$, $\mathcal{H}(x)$, is a subspace of
$H^{1}(X,Z(\omega),\mathbb{R})$ so that the tangent space to the
$\mathrm{SL}(2,\mathbb{R}$-orbit of $(X,\omega)$ is contained in
$\mathcal{H}(x)\otimes\mathbb{C}$.

######  Definition 2.3.

The complex vector bundle $H^{1}$ has a symplectic intersection form given by

$\langle{\omega_{1},\omega_{2}\rangle}:=\frac{i}{2}\int_{X}\omega_{1}\wedge\overline{\omega_{2}}$

for $\omega_{1},\omega_{2}$ in a fiber of $H^{1}.$ Henceforth, we will refer
to a tangent vector $v\in H^{1}(X,Z(\omega),\mathbb{C})$ at a point
$(X,\omega)$ as in a direction tranversal to the
$\mathrm{SL}(2,\mathbb{R})$-orbit of $(X,\omega)$ if $p(v)$ is in the
symplectic complement of the image under $p$ of the tangent space to the
$\mathrm{SL}(2,\mathbb{R})$-orbit, $p(\mathbf{H})\subset
p(\mathcal{H}(x)\otimes_{\mathbb{R}}\mathbb{C})$. For any such $v\in
H^{1}(X,Z(\omega),\mathbb{C})$ in this complement, we write $v\in
p(\mathbf{H})^{\perp}$.

By the preceeding discussion, we have a decomposition of the tangent bundle

$T_{\mathbb{C}}(\tilde{\mathcal{N}})=T_{\mathbb{C}}^{st}(\tilde{N})\oplus
T_{\mathbb{C}}^{bal}(\tilde{\mathcal{N}})$ (2.3)

where $T_{\mathbb{C}}(\tilde{\mathcal{N}})^{bal}$ consists of the elements
$v\in T_{\mathbb{C}}(\tilde{\mathcal{N}})$ so that $v\in
p(\mathbf{H})^{\perp}$, using the notation in Definition 2.3. This
decomposition is invariant under the action of $\mathrm{SL}(2,\mathbb{R})$,
but is not invariant under parallel transport unless $\tilde{\mathcal{N}}$ is
a Teichmüller curve.

######  Definition 2.4 (AGY-metric, [4]).

Let $x=(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$. Identify the tangent
space at $(X,\omega)$ with $H^{1}(X,Z(\omega),\mathbb{C})$. For a tangent
vector $v\in H^{1}(X,Z(\omega),\mathbb{C})$, consider the norm

$\lvert\lvert
v\rvert\rvert_{x}:=\underset{\mathbf{s}\in\mathcal{S}}{\mathrm{sup}}\Big{\lvert}\frac{v(\mathbf{s})}{\mathrm{Hol}_{\omega}(\mathbf{s})}\Big{\rvert}$

where $\mathcal{S}$ denotes the set of saddle connections on $(X,\omega)$, and
$\mathrm{Hol}_{\omega}(\mathbf{s})\in\mathbb{C}$ is the holonomy vector
associated to $\mathbf{s}$ for the form $\omega$. The set of holonomy vectors
is always discrete. It is shown in [4] that this definition induces a complete
Finsler metric on $\Omega_{1}\mathcal{M}_{g}(\sigma)$. Let
$\kappa:[0,1]\rightarrow\Omega_{1}\mathcal{M}_{g}(\sigma)$ be a differentiable
path and denote its AGY-length by

$l(\kappa):=\int_{0}^{1}\lvert\lvert\kappa^{\prime}(t)\rvert\rvert_{\kappa(t)}\
dt.$

The distance function associated to this length will be denoted by $d_{AGY}$.

######  Lemma 2.5 ([5]).

Let $\kappa:[0,1]\rightarrow\Omega_{1}\mathcal{M}_{g}(\sigma)$ be a
differentiable path. Then for any tangent vector $v\in
H^{1}(X,Z(\omega),\mathbb{C})$,

$e^{-\mathrm{length}(\kappa)}\leq\frac{\lvert\lvert
v\rvert\rvert_{\kappa(0)}}{\lvert\lvert v\rvert\rvert_{\kappa(1)}}\leq
e^{\mathrm{length}(\kappa)}.$

The following and its proof are essentially Proposition 5.3 in [5]; there, it
is stated for vectors in the unstable space, but the proof goes through for
any choice of tangent vectors. We briefly recapitulate the proof ingredients
for the general case, as we will make extensive use of the AGY-norm.

######  Lemma 2.6 ("Exponential map", [5]).

Let $x=(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$ and $v\in
H^{1}(X,Z(\omega),\mathbb{C})$, a tangent vector. Consider the path
$\kappa:[0,1]\rightarrow\Omega_{1}\mathcal{M}_{g}(\sigma)$ starting from $x$
satisfying $\kappa^{\prime}(t)=v$ for all $t\in[0,1]$. Implicitly, we assume
$\kappa(t)$ is well-defined for all $t\in[0,1]$. Define
$\mathrm{exp}_{x}(v):=\kappa(1)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$. Let
$B(0,r)$ denote the ball of radius $r$ centered at the origin of
$H^{1}(X,Z(\omega),\mathbb{C})$ with respect to the norm
$\lvert\lvert\cdot\rvert\rvert_{x}$. Then, there exists an absolute constant
$C_{1}>0$ and $C_{2}=(C_{1}+1)^{2}C_{1}$ such that the following hold:

  1. 1.

for the rescaling $\omega\mapsto s\omega$, $s\in\mathbb{R}$,
$l(\kappa)=l(s\kappa)$. That is, the AGY-metric is invariant under real
scalings.

  2. 2.

The exponential map
$\mathrm{exp}_{x}:H^{1}(X,Z(\omega),\mathbb{C})\rightarrow\Omega_{1}\mathcal{M}_{g}(\sigma)$
is well-defined over $B(0,1/C_{1})$.

  3. 3.

For every $v\in B(0,1/C_{1})$,

$d_{AGY}(x,\mathrm{exp}_{x}(v))\leq C_{1}\lvert\lvert v\rvert\rvert_{x}.$

  4. 4.

For every $v\in B(0,1/C_{1})$ and every $w\in H^{1}(X,Z(\omega),\mathbb{C})$,

$1/C_{1}\leq\frac{\lvert\lvert w\rvert\rvert_{x}}{\lvert\lvert
w\rvert\rvert_{\mathrm{exp}_{x}(v)}}\leq C_{1}.$

  5. 5.

For $v\in B(0,1/C_{2})$,

$d_{AGY}(x,\mathrm{exp}_{x}(v))\geq\lvert\lvert v\rvert\rvert_{\omega}/C_{1}.$

###### Proof.

For 1., we have

$\displaystyle l(s\kappa)$ $\displaystyle=\int_{0}^{1}\lvert\lvert
s\kappa^{\prime}(t)\rvert\rvert_{s\kappa(t)}\ dt$
$\displaystyle=\int_{0}^{1}\underset{\mathbf{s}\in\mathcal{S}}{\mathrm{sup}}\Bigg{\lvert}\frac{s\kappa^{\prime}(t)(\mathbf{s})}{\mathrm{Hol}_{s\kappa(t)}(\mathbf{s})}\Bigg{\rvert}dt$
$\displaystyle=\int_{0}^{1}\underset{\mathbf{s}\in\mathcal{S}}{\mathrm{sup}}\Bigg{\lvert}\frac{s\kappa^{\prime}(t)(\mathbf{s})}{s\mathrm{Hol}_{\kappa(t)}(\mathbf{s})}\Bigg{\rvert}dt$
$\displaystyle=l(\kappa).$

For 2., let $\kappa:[0,1]\rightarrow\Omega_{1}\mathcal{M}_{g}(\sigma)$ be a
path with $\kappa(0)=(X,\omega)$ and $\kappa^{\prime}(t)=v\ \forall
t\in[0,1]$. By Lemma 2.5, we have

$\lvert\lvert\kappa^{\prime}(t)\rvert\rvert_{\kappa(t)}=\lvert\lvert
v\rvert\rvert_{x}e^{\int_{0}^{t}\lvert\lvert\kappa^{\prime}(r)\rvert\rvert_{\kappa(r)}\
dr}.$ (2.4)

Consider the length function $L(t):\mathbb{R}\mapsto\mathbb{R}$ defined by
$L(t)=\int_{0}^{t}\lvert\lvert\kappa^{\prime}(r)\rvert\rvert_{\kappa(r)}\ dr$.
By Equation 2.4, we have $L^{\prime}(t)\leq e^{L(t)}\lvert\lvert
v\rvert\rvert_{x}.$ Integrating both sides of this inequality and rearranging,
we have

$\lvert\lvert\kappa^{\prime}(t)\rvert\rvert_{\kappa(t)}\leq\frac{\lvert\lvert
v\rvert\rvert_{x}}{1-t\lvert\lvert v\rvert\rvert_{x}},$

which immediately implies that for any $C_{1}>1$, $\mathrm{exp}_{\omega}(v)$
is well-defined on the ball $B(0,1/C_{1})$. Further, note that

$d_{AGY}(x,\mathrm{exp}_{x}(v))\leq\int_{0}^{1}\lvert\lvert\kappa^{\prime}(t)\rvert\rvert_{\kappa(t)}\leq\frac{\lvert\lvert
v\rvert\rvert_{x}}{(1-\lvert\lvert v\rvert\rvert_{x})},$

which implies $d_{AGY}(x,\mathrm{exp}_{x}(v))\leq C_{1}\lvert\lvert
v\rvert\rvert_{x}$ for $v\in B(0,1/C_{1})$, establishing 3. Since
$\mathrm{exp}_{\omega}$ is well-defined over $B(0,1/C_{1})$ by 2., another
application of Lemma 2.5 gives

$e^{-L(1)}\leq\frac{\lvert\lvert w\rvert\rvert_{x}}{\lvert\lvert
w\rvert\rvert_{\mathrm{exp}_{x}(v)}}\leq e^{L(1)}$

for any $w\in H^{1}(X,Z(\omega),\mathbb{C})$, proving 4., since
$L(1)\leq\mathrm{log}(C_{1})$. Finally, we claim that for
$C_{2}=(C_{1}+1)^{2}C_{1}$, we have

$d_{AGY}(x,\mathrm{exp}_{x}(v))\geq\lvert\lvert v\rvert\rvert_{x}/C_{1}$

so long as $v\in B(0,1/C_{2})$. To see this, consider a (nearly) length-
minimzing path $\kappa$ connecting $(X,\omega)$ and
$\mathrm{exp}_{\omega}(v)$. By part 3., we have

$l(\kappa)\leq C_{1}\lvert\lvert
v\rvert\rvert_{x}\leq\frac{1}{(C_{1}+1)^{2}}.$ (2.5)

Consider a lift of $\kappa$, $\tilde{\kappa}$ to the tangent space to the
stratum identified with $H^{1}(X,Z(\omega),\mathbb{C})$. By part 4., we have
$\lvert\lvert\tilde{\kappa}^{\prime}(t)\rvert\rvert_{x}\leq
C_{1}\lvert\lvert\tilde{\kappa}^{\prime}(t)\rvert\rvert_{\kappa(t)}$. This
inequality can be integrated from $0$ to $t$ to obtain

$\lvert\lvert\tilde{\kappa}(t)\rvert\rvert_{x}\leq C_{1}l(\kappa)\leq
C_{1}\frac{1}{(C_{1}+1)^{2}}<1/C_{1}.$ (2.6)

We conclude that $\tilde{\kappa}(t)\in B(0,1/C_{1})\ \forall t\in[0,1]$ and in
particular $\tilde{\kappa}(1)$ is well-defined and we have

$\displaystyle\lvert\lvert\tilde{\kappa}(1)\rvert\rvert_{x}=\lvert\lvert
v\rvert\rvert_{x}$ $\displaystyle\leq C_{1}l(\kappa)$ $\displaystyle\leq
C_{1}d_{AGY}(x,\mathrm{exp}_{x}(v)),$

proving 5. ∎

###  2.5 Unstable and stable foliations

In contrast to the homogeneous setting, the unstable and stable foliations for
the geodesic flow are only locally defined. We review these definitions and
their relationships to our objects of interest.

######  Definition 2.7 (Proposition 4.1 [5]).

Recall that for any $x=(X,\omega)\in\Omega\mathcal{M}_{g}(\sigma)$ there is an
open neighborhood $U\subset\Omega\mathcal{M}_{g}(\sigma)$ containing it so
that the period map $\phi_{T,\omega}:U\rightarrow
H^{1}(X,Z(\omega),\mathbb{C})\cong\mathbb{C}^{2g+Z(\omega)-1}$ is an open
analytic embedding. Henceforth, we suppress the notation for the triangulation
$T$ and write $\phi_{\omega}$. Further, recall that the tangent space to
$(X,\omega)$ in the stratum can be identified with
$T_{\omega}\Omega\mathcal{M}_{g}(\sigma)\cong
H^{1}(X,Z(\omega),\mathbb{C})\cong
H^{1}(X,Z(\omega),\mathbb{R})\otimes_{\mathbb{R}}\mathbb{C}\cong
H^{1}(X,Z(\omega),\mathbb{R})\oplus H^{1}(X,Z(\omega),i\mathbb{R})$. Then,
there is a decomposition of the tangent space

$T_{x}\Omega\mathcal{M}_{g}(\sigma)\cong\mathbf{v}(x)\oplus E^{u}(x)\oplus
E^{s}(x)$

where $\mathbf{v}(x)$ is the direction of the Teichmüller geodesic flow, and

$\displaystyle E^{u}(x)=$ $\displaystyle
T_{x}\Omega\mathcal{M}_{g}(\sigma)\cap
D\phi_{\omega}^{-1}(H^{1}(X,Z(\omega),\mathbb{R})),$ $\displaystyle E^{s}(x)=$
$\displaystyle T_{x}\Omega\mathcal{M}_{g}(\sigma)\cap
D\phi_{\omega}^{-1}(H^{1}(X,Z(\omega),i\mathbb{R})).$

These subspaces are integrable, and depend smoothly on $(X,\omega)$. We will
refer to the integral leaves of $E^{u}(x)$ and $E^{s}(x)$, respectively, as
the unstable and stable manifolds $W^{u}(x)$ and $W^{s}(x)$. These are affine
submanifolds of $\Omega\mathcal{M}_{g}(\sigma)$ in the sense of Definition
2.1,

It follows from Lemma 3.1 in [41] that the foliations $W^{u}(x)$ and
$W^{s}(x)$ are well-defined on the unit-area locus
$\Omega_{1}\mathcal{M}_{g}(\sigma)$.

######  Definition 2.8 (Period boxes).

Let $(\tilde{X},\tilde{\omega})\in\Omega_{1}\mathcal{T}_{g}(\sigma)$. Define,
for every $r>0$,

$R_{r}(\tilde{X},\tilde{\omega}):=\left\\{\phi_{\omega}(\tilde{x},\tilde{\omega})+u+iv\
:\ u,v\in H^{1}(X,Z(\omega),\mathbb{R}\ ,\lvert\lvert u+iv\rvert\rvert\leq
r\right\\}.$

where we are again using the identification
$H^{1}(X,Z(\omega),\mathbb{C})\cong H^{1}(X,Z(\omega),\mathbb{R}\oplus
H^{1}(X,Z(\omega),i\mathbb{R})$.

Note that we may choose $r>0$ small enough so that

$\phi^{-1}_{\omega}:R_{r}(\tilde{X},\tilde{\omega})\cap\phi_{\omega}(\Omega_{1}\mathcal{T}_{g}(\sigma))\mapsto\Omega_{1}\mathcal{T}_{g}(\sigma)$

is a homeomorphism. Set
$B_{r}(\tilde{X},\tilde{\omega})=\phi^{-1}_{\omega}\left(R_{r}(\tilde{X},\tilde{\omega})\right)$.
Observe that, by Lemma 2.6, $B_{r}(\tilde{X},\tilde{\omega})$ is well-defined
for all $0<r<1/C_{1}$ and all
$(\tilde{X},\tilde{\omega})\in\Omega_{1}\mathcal{T}_{g}(\sigma)$. Further, for
any $(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$, there exists
$0<r(\omega)<1/C_{1}$ so that the restriction of the projection map from the
Teichmüller space to moduli space
$\pi|_{B_{r(\omega)}(\tilde{X},\tilde{\omega})}:\Omega_{1}\mathcal{T}_{g}(\sigma)\rightarrow\Omega_{1}\mathcal{M}_{g}(\sigma)$
is injective. For all $t,s\in\mathbb{R}$, let
$a_{t}=\begin{bmatrix}e^{t/2}&0\\\ 0&e^{-t/2}\end{bmatrix}$ and
$u_{s}=\begin{bmatrix}1&s\\\ 0&1\end{bmatrix}$. The action of the transpose of
$u_{s}$ is given by $u_{s}^{\intercal}=\begin{bmatrix}1&0\\\
s&1\end{bmatrix}$. Recall that the two one-parameter subgroups $a_{t}$ and
$u_{s}$ of $\mathrm{SL}(2,\mathbb{R})$ define, respectively, the Teichmüller
geodesic flow, and horocycle flow on $\Omega_{1}\mathcal{M}_{g}(\sigma)$. The
horocycle flow $u_{s}$ preserves the leaves of the unstable foliation
$W^{u}(\omega)$ and the tranpose $u_{s}^{\intercal}$ preserves the leaves of
the stable foliation $W^{s}(\omega)$. To see this explicitly, let
$z=\begin{bmatrix}x_{1}&...&x_{n}\\\ y_{1}&...&y_{n}\end{bmatrix}$, and
$z^{\prime}=\begin{bmatrix}x_{1}^{\prime}&...&x_{n}^{\prime}\\\
y_{1}^{\prime}&...&y_{n}^{\prime}\end{bmatrix}$ be two points in the same
local period coordinate chart. If we write $\phi_{\omega}(X,\omega)=u+iv$, the
stable leaves of $W^{s}(x)$ are locally identified with
$\phi_{\omega}(X,\omega)+iw$ and $w$ is some row $n$-vector, by Definition
2.7. Therefore, we see that $z$ and $z^{\prime}$ are in the same stable leaf
if

$\begin{bmatrix}x_{1}&...&x_{n}\\\
y_{1}&...&y_{n}\end{bmatrix}=\begin{bmatrix}x_{1}^{\prime}&...&x_{n}^{\prime}\\\
y_{1}^{\prime}&...&y_{n}^{\prime}\end{bmatrix}+\begin{bmatrix}0&...&0\\\
w_{1}&...&w_{n}\end{bmatrix}.$

The analogous observation is true for the unstable leaves. Let
$e^{-0.01t}<\beta<1$. For every $t\geq 0$, define

$E_{t}=B_{\beta}^{s}\cdot
a_{t}\cdot\left\\{u_{r}:r\in[0,1]\right\\}\subset\mathrm{SL}(2,\mathbb{R})$

where
$B_{\beta}:=\left\\{u_{s}^{\intercal}:|s|\leq\beta\right\\}\cdot\left\\{a_{t}:|t|\leq\beta\right\\}$.
In words, this is a "smearing" by a small amount of
$\mathrm{SL}(2,\mathbb{R})$ of a geodesic push of length $t$ of a length-1
horocycle segment. As we have just observed, this is in fact akin to a
Margulis thickening in the unstable direction, but only along the
$\mathrm{SL}(2,\mathbb{R})$-orbit, rather than the ambient space. We recall
that for any such $E_{t}$ as above, and any
$(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$, we define a map
$E_{t}\rightarrow\Omega_{1}\mathcal{M}_{g}(\sigma)$ given by $h\mapsto
h\cdot(X,\omega)$, since any $h\in E_{t}$ is in particular an element of
$\mathrm{SL}(2,\mathbb{R})$. We see that this map is injective if, for any
lift of $(X,\omega)$, $(\tilde{X},\tilde{\omega})$, the restricted projection
map $\pi|_{B_{r}(\tilde{X},\tilde{\omega})\cap
E_{t}\cdot(\tilde{X},\tilde{\omega})}:\Omega_{1}\mathcal{T}_{g}(\sigma)\rightarrow\Omega_{1}\mathcal{M}_{g}(\sigma)$
is injective.

##  3 Quantitative non-divergence

We collect quantitative non-divergence results in a form convenient for our
application.

######  Definition 3.1 (Injectivity radius).

For $(X,\omega)\in\Omega_{1}\mathcal{T}_{2}(2)$, let $\mathcal{S}_{\omega}$
denote the set of saddle connections of $\omega$ on $X$. We will use
$l_{\omega,\mathbf{s}}=\lvert\mathrm{Hol}_{\omega}(\mathbf{s})\rvert$ to
denote the length of $\mathbf{s}$ in the conical flat metric $|\omega|$.
Recall that we have a quotient map
$\pi:\Omega_{1}\mathcal{T}_{2}(2)\rightarrow\Omega_{1}\mathcal{M}_{2}(2)$. We
define the injectivity radius of $(X,\omega)\in\Omega_{1}\mathcal{T}_{2}(2)$,
denoted by $\mathrm{inj}(X,\omega)$ to be the infinum

$\mathrm{inj}(X,\omega)=\underset{s\in\mathcal{S}_{\omega}}{\mathrm{inf}}\lvert\mathrm{Hol}_{\omega}(\mathbf{s})\rvert.$

Define for $\epsilon>0$,

$\Omega_{1}\mathcal{M}_{\epsilon}=\pi\left(\left\\{(X,\omega)\in\Omega_{1}\mathcal{T}_{2}(2)\
|\ \mathrm{inj}(X,\omega)\geq\epsilon\right\\}\right).$

If $(X,\omega)\in\Omega_{1}\mathcal{M}_{\epsilon}$ (resp.
$(X,\omega)\notin\Omega_{1}\mathcal{M}_{\epsilon})$, we will also say
$\mathrm{inj}(X,\omega)\geq\epsilon$ (resp.
$\mathrm{inj}(X,\omega)<\epsilon)$.

######  Theorem 3.2 (Minsky-Weiss [34]).

There are positive constants $\kappa_{1}$, $\kappa_{2}$, $\alpha$ ($\alpha\leq
1)$ depending only on the stratum $\Omega_{1}\mathcal{M}_{g}(\sigma)$ such
that if every $(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$,
$s\in\mathcal{S}_{\omega}$, an interval $I\subset\mathbb{R}$, and
$\rho^{\prime}>0$ satisfy:

$\underset{s\in I}{\mathrm{sup}}\
l_{u_{s}\omega,\mathbf{s}}\geq\rho^{\prime},$

then for any $0<\epsilon<\kappa_{1}\rho^{\prime}$ we have

${\big{|}\left\\{r\in
I:\mathrm{inj}(u_{s}(X,\omega))<\epsilon\right\\}\big{|}}<\kappa_{2}\cdot\left(\frac{\epsilon}{\rho^{\prime}}\right)^{\alpha}|I|.$

For clarity, we note that the condition $\underset{s\in I}{\mathrm{sup}}\
l_{u_{s}\omega,\mathbf{s}}\geq\rho^{\prime}$ just means that there is no
saddle connection on the surface $(X,\omega)$ whose length in the conical flat
metric remains below $\rho^{\prime}$ over a time-length $|I|$ flow from
$(X,\omega)$. A consequence of the arguments in this section is a proof of the
following lemma.

######  Lemma 3.3.

There exist constants $\alpha,C_{3},C_{4}>0$ (depending only on the stratum)
so that the following property holds. Let $0<\epsilon,\eta<1$ and
$(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma).$ Let $I\subset[-10,10]$ be an
interval and assume $|I|\geq\eta$. Then,

${\big{|}\left\\{s\in
I:\mathrm{inj}\left(a_{t}u_{s}(X,\omega)\right)<{\epsilon}\right\\}\big{|}}<C_{4}{\epsilon}^{\alpha}|I|,$

so long as $t\geq\big{\lvert}\mathrm{log}\left(\eta\
\mathrm{inj}(X,\omega)\right)^{-1}\big{\rvert}+C_{3}$. The constants are
explicit; $\alpha$ is the $\alpha$ appearing in Theorem 3.2,
$C_{3}=\mathrm{log}\left(\frac{1}{\kappa_{1}}\right)+\mathrm{log}(\sqrt{2})$,
and $C_{4}=\frac{\kappa_{2}}{\kappa_{1}^{\alpha}}$ where
$\kappa_{1},\kappa_{2}$ are the constants appearing in Theorem 3.2.

Although we only require Lemma 3.3 for subsequent use in this paper, we record
an auxiliary statement, to emphasize the analogy to quantitative non-
divergence results of this kind in the homogeneous setting. We refer the
reader to Section 3 of [26], where the corresponding statements are proven for
the cases of $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ and
$\mathrm{SL}(2,\mathbb{C})$. We recall the following definitions due to
Minsky-Weiss.

######  Definition 3.4 (Good functions, Definition 3.1 [34]).

Let $\mathcal{F}$ be a collection of continuous functions
$\mathbb{R}\rightarrow\mathbb{R}_{+}$, and $I\subset\mathbb{R}$ some interval.
For $\theta>0$ and $f\in\mathcal{F}$, make the following definitions:

$\displaystyle I_{f,\theta}$ $\displaystyle=\left\\{s\in
I:f(s)<\theta\right\\}$ $\displaystyle I_{\mathcal{F},\theta}$
$\displaystyle=\left\\{s\in I:\exists f\in\mathcal{F},\ f(s)<\theta\right\\}$
$\displaystyle\lvert\lvert f\rvert\rvert_{I}$ $\displaystyle=\underset{s\in
I}{\mathrm{sup}}\ f(s).$

Let $\kappa,\alpha,\rho>0$. We designate $\mathcal{F}$ as
$(\kappa,\alpha,\rho)$-good if it satisfies the following property. For any
interval $I\subset\mathbb{R}$ and any $f\in\mathcal{F}$, we have, for
$0<\epsilon<\rho$,

$\frac{\big{\lvert}I_{f,\epsilon}\big{\rvert}}{\lvert
I\rvert}\leq\kappa\left(\frac{\epsilon}{\lvert\lvert
f\rvert\rvert_{I}}\right)^{\alpha}.$

We call $\mathcal{F}$ $(\kappa,\alpha)$-good if it is
$(\kappa,\alpha,\rho)$-good for all $\rho$.

######  Definition 3.5 (Sparse-covering property).

We say a pair consisting of a collection of continuous functions $\mathcal{F}$
and an interval $I$ as above, $(\mathcal{F},I)$, satisfies the sparse-covering
property if the following conditions hold. There exist
$\kappa,\alpha,\rho,M>0$ such that

  1. 1.

$\mathcal{F}$ is $(\kappa,\alpha,\rho)$-good.

  2. 2.

For every $f\in\mathcal{F}$, $\lvert\lvert f\rvert\rvert_{I}\geq\rho$.

  3. 3.

For every $s\in I$,

$\\#\left\\{f\in\mathcal{F}:f(s)<\rho\right\\}\leq M.$

######  Proposition 3.6.

Let $(F,I)$ satisfy the sparse-covering property in the sense of Definition
3.5 for the constants $\kappa,\alpha,\rho,M>0$. Then, for every
$0<\epsilon<\rho$,

$\frac{\big{\lvert}I_{\mathcal{F},\epsilon}\big{\rvert}}{\lvert
I\rvert}\leq\kappa M\left(\frac{\epsilon}{\lvert\lvert
f\rvert\rvert_{I}}\right)^{\alpha}.$

######  Proposition 3.7 ([34]).

For any $(X,\omega)\in\Omega_{1}\mathcal{T}_{g}(\sigma)$, and
$\mathbf{s}\in\mathcal{S}_{\omega}$, define a function
$l_{\omega,\mathbf{s}}(s):\mathbb{R}\rightarrow\mathbb{R}_{+}$ by
$l_{\omega,\mathbf{s}}(s)=l_{u_{s}\omega,\mathbf{s}}$. That is, for a point in
moduli space, and a saddle connection on the surface, consider the function
measuring the length of the saddle connection along a horocycle segment.
Consider the collection of functions

$\mathcal{F}=\left\\{l_{\omega,\mathbf{s}}:(X,\omega)\in\Omega_{1}\mathcal{T}_{g}(\sigma),\mathbf{s}\in\mathcal{S}_{\omega}\right\\}.$

$\mathcal{F}$ is $(2,1)$-good in the sense of Definition 3.4.

Unfortunately, there are $I\subset\mathbb{R}$ for which $(\mathcal{F},I)$
doesn’t satisfy the sparse-covering property. However, the arguments in
Section 6 of Minsky-Weiss [34] show:

######  Lemma 3.8.

There exists a subset
$\mathcal{F}_{0}\subset\mathcal{F}=\left\\{l_{\omega,\mathbf{s}}:(X,\omega)\in\Omega_{1}\mathcal{T}_{g}(\sigma),\mathbf{s}\in\mathcal{S}_{\omega}\right\\}.$
so that $(\mathcal{F}_{0},I)$ satisfies the sparse-covering property for any
$I\subset\mathbb{R}$ and for which the following property holds. There are
positive constants $\kappa_{1}$, $\kappa_{2}$, $\alpha$ depending only on the
stratum $\Omega_{1}\mathcal{M}_{g}(\sigma)$ such that if every
$f(s)\in\mathcal{F}_{0}$, an interval $I\subset\mathbb{R}$, and
$\rho^{\prime}>0$ satisfy:

$\underset{s\in I}{\mathrm{sup}}\ f(s)\geq\rho^{\prime},$

then for any $0<\epsilon<\kappa_{1}\rho^{\prime}$ we have

${\big{|}\left\\{s\in
I:\mathrm{inj}(u_{s}(X,\omega))<\epsilon\right\\}\big{|}}<\kappa_{2}\cdot\left(\frac{\epsilon}{\rho^{\prime}}\right)^{\alpha}|I|.$

######  Lemma 3.9.

There exists a subset
$\mathcal{F}_{0}\subset\mathcal{F}=\left\\{l_{\omega,\mathbf{s}}:(X,\omega)\in\Omega_{1}\mathcal{T}_{g}(\sigma),\mathbf{s}\in\mathcal{S}_{\omega}\right\\}.$
so that $(\mathcal{F}_{0},I)$ satisfies the sparse-covering property for any
$I\subset\mathbb{R}$ and for which the following property holds. There are
positive constants $\kappa_{1}$, $\kappa_{2}$, $\alpha$ depending only on the
stratum $\Omega_{1}\mathcal{M}_{g}(\sigma)$ such that if every
$f(s)\in\mathcal{F}_{0}$, a symmetric interval $I\subset\mathbb{R}$ with
$\lvert I\rvert>\eta$, and $\rho^{\prime}>0$ satisfy:

$\underset{f\in\mathcal{F}_{0}}{\mathrm{inf}}\ f(0)\geq\rho^{\prime},$

then for any $0<\epsilon,\eta<1$ we have

${\big{|}\left\\{r\in
I:\mathrm{inj}(a_{t}u_{s}(X,\omega))<\epsilon\right\\}\big{|}}<\kappa_{2}\cdot\left({\epsilon}\right)^{\alpha}|I|,$

provided
$t\geq\mathrm{max}\left\\{\Big{\lvert}\mathrm{log}\left(\sqrt{\frac{\rho^{\prime}}{2}}\right)^{-1}\Big{\rvert},\Big{\lvert}\mathrm{log}\
\left(\eta\sqrt{\frac{\rho^{\prime}}{2}}\right)^{-1}\Big{\rvert}\right\\}$.

###### Proof.

We first note that since $\underset{f\in\mathcal{F}_{0}}{\mathrm{inf}}\
f(0)\geq\rho^{\prime}$, we have $\underset{f\in\mathcal{F}_{0}}{\mathrm{sup}}\
f(0)\geq\rho^{\prime}$. That is, the longest saddle connection in the family
$\mathcal{F}_{0}$ on $(X,\omega)$ is of length greater than $\rho^{\prime}$.
Let $\lvert\mathrm{Re}_{\omega}(\mathbf{s})\rvert$,
$\lvert\mathrm{Im}_{\omega}(\mathbf{s})\rvert$, denote, respectively the
horizontal and vertical lengths of $s$ with respect to $\omega$. Then,

$\displaystyle\underset{r\in I}{\mathrm{sup}}\
\lvert\mathrm{Re}_{a_{t}u_{r}\omega}(\mathbf{s})\rvert$ $\displaystyle\geq
e^{t}\lvert\mathrm{Re}_{\omega}(\mathbf{s})\rvert.$ (3.1)

Since $\lvert\mathrm{Hol}_{\omega}(\mathbf{s})\rvert\geq\rho^{\prime}$,
$\mathrm{max}\left\\{\lvert\mathrm{Re}_{\omega}(\mathbf{s})\rvert,\lvert\mathrm{Im}_{\omega}(\mathbf{s})\rvert\right\\}\geq\sqrt{\frac{\rho^{\prime}}{2}}$.
Suppose
$\lvert\mathrm{Re}_{\omega}(\mathbf{s})\rvert\geq\sqrt{\frac{\rho^{\prime}}{2}}$.
Then, we have,

$\displaystyle\underset{s\in I}{\mathrm{sup}}\
l_{a_{t}u_{s}\omega,\mathbf{s}}\geq\underset{s\in I}{\mathrm{sup}}\
\lvert\mathrm{Re}_{a_{t}u_{s}\omega}(\mathbf{s})\rvert$ $\displaystyle\geq
e^{t}\lvert\mathrm{Re}_{\omega}(\mathbf{s})\rvert$ (3.2) $\displaystyle\geq
e^{t}\sqrt{\frac{\rho^{\prime}}{2}}.$ (3.3)

Hence, $\underset{s\in I}{\mathrm{sup}}\
l_{a_{t}u_{s}\omega,\mathbf{s}}\geq\frac{1}{\kappa_{1}}$, so long as
$t\geq\Big{\lvert}\left(\mathrm{log}\sqrt{\frac{\rho^{\prime}}{2}}\right)^{-1}\Big{\rvert}+\big{\lvert}\mathrm{log}\left(\frac{1}{\kappa_{1}}\right)\big{\rvert}$.
On the other hand, suppose
$\lvert\mathrm{Im}_{\omega}(\mathbf{s})\rvert\geq\sqrt{\frac{\rho^{\prime}}{2}}$.
Then,

$\displaystyle\underset{s\in I}{\mathrm{sup}}\
\lvert\mathrm{Im}_{a_{t}u_{s}\omega}(\mathbf{s})\rvert$ $\displaystyle\geq
e^{t}\eta\lvert\mathrm{Im}_{\omega}(\mathbf{s})\rvert.$ (3.4)

Therefore, $\underset{s\in I}{\mathrm{sup}}\
l_{a_{t}u_{s}\omega,\mathbf{s}}\geq\frac{1}{\kappa_{1}}$, so far as
$t\geq\Big{\lvert}\left(\mathrm{log}\
\eta\sqrt{\frac{\rho^{\prime}}{2}}\right)^{-1}\Big{\rvert}+\big{\lvert}\mathrm{log}\left(\frac{1}{\kappa_{1}}\right)\big{\rvert}$.
By Lemma 3.8, and the fact that $a_{t}u_{s}=u_{se^{2t}}a_{t}$, we have

$\frac{{\big{|}\left\\{s\in
I:\mathrm{inj}(a_{t}u_{s}(X,\omega))<\epsilon\right\\}\big{|}}}{\lvert
I\rvert}=\frac{{\big{|}\left\\{r\in
I^{\prime}:\mathrm{inj}(u_{s}a_{t}(X,\omega))<\epsilon\right\\}\big{|}}}{\lvert
I^{\prime}\rvert}<\frac{\kappa_{2}}{\kappa_{1}^{\alpha}}\cdot\left({\epsilon}\right)^{\alpha},$

where $I^{\prime}:=e^{2t}I$, given that
$t\geq\Big{\lvert}\left(\mathrm{log}\sqrt{\frac{\rho^{\prime}}{2}}\right)^{-1}\Big{\rvert}+\big{\lvert}\mathrm{log}\left(\frac{1}{\kappa_{1}}\right)\big{\rvert}$
(recall $\eta$ < 1). ∎

###### Proof of Lemma 3.3.

Note that, by the definition of injectivity radius, if
$\mathrm{inj}(X,\omega)\geq\rho^{\prime}$, then using the notation of Lemma
3.9, $\underset{f\in\mathcal{F}_{0}}{\mathrm{inf}}\ f(0)\geq\rho^{\prime}$.
Lemma 3.3 follows by the expressions given for the explicit constants $C_{3}$
and $C_{4}$ in the statement of the Lemma. ∎

##  4 Margulis function

###  4.1 Non-divergence and injectivity estimates

We combine some results in the literature into a proposition quantifying the
injectivity of the projection map from Teichmüller space to moduli space as a
point varies along an orbit.

######  Theorem 4.1 ([10],[3]).

There exists a continuous, proper, $\mathrm{SO}(2)$-invariant function
$u:\Omega_{1}\mathcal{M}_{g}(\sigma)\rightarrow[2,\infty)$ such that there
exists a constant $\kappa_{3}$ depending only on the stratum so that for all
$x=(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$ and all $t>0$,

$e^{-\kappa_{3}t}u(x)\leq u(a_{t}x)\leq e^{\kappa_{3}t}u(x).$ (4.1)

Recall that for $\tilde{x}=(\tilde{X},\tilde{\omega})$ we defined in
Subsection 2.5 the restricted projection map
$\pi|_{B_{r}(\tilde{X},\tilde{\omega})}:\Omega_{1}\mathcal{T}_{g}(\sigma)\rightarrow\Omega_{1}\mathcal{M}_{g}(\sigma)$.
By non-divergence statements close to those in Section 3, the following is
proven in [13].

######  Lemma 4.2 (Lemma 2.6, [13]).

There exists a constant $C_{5}>0$ depending only on the stratum so that for
all $\tilde{x}=(\tilde{X},\tilde{\omega})\in\Omega_{1}\mathcal{T}_{g}(\sigma)$
and every $0<r\leq u(x)^{-C_{5}}$ the restricted projection map
$\pi|_{B_{r}(\tilde{x})}$ is injective.

As a consequence, we have the following proposition.

######  Proposition 4.3.

For any $t>0$ and any
$\tilde{x}=(\tilde{X},\tilde{\omega})\in\Omega_{1}\mathcal{T}_{g}(\sigma)$,
the restricted projection map $\pi|_{B_{r}({a_{t}\tilde{x}})}$ is injective
for $r<e^{-C_{5}\kappa_{3}t}u(x)$.

###  4.2 Non-uniformly hyperbolic dynamics of the geodesic flow

Recall the Kontsevich-Zorich cocyle defined in Subsection 2.4,
$A:\mathrm{SL}(2,\mathbb{R})\times\Omega_{1}\mathcal{M}_{g}(\sigma)\rightarrow\mathrm{Sp}(2g,\mathbb{Z})\ltimes\mathbb{R}^{h}$.
For $x=(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$ and $v\in
H^{1}(X,\mathbb{R})$, its Lyapunov exponents $\lambda_{i}$ are

$\lambda_{i}=\lim\limits_{t\rightarrow\infty}\frac{1}{t}\mathrm{log}\frac{||A(a_{t},x)v||}{||v||}.$

For $\mu_{MV}$ the Masur-Veech measure on a connected component $\mathcal{C}$
of a stratum $\Omega_{1}\mathcal{M}_{g}(\sigma)$, a foundational result of
Forni [17] gives

######  Theorem 4.4.

The Lyapunov exponents of the Kontsevich-Zorich cocycle for $\mu_{MV}$ are all
non-zero and distinct.

Theorem 4.4 implies non-uniform hyperbolicity for the geodesic flow for the
Hodge norm, which we will now define. Recall the inner product from Definition
2.3 given by
$\langle{\omega_{1},\omega_{2}\rangle}:=\frac{i}{2}\int_{X}\omega_{1}\wedge\overline{\omega_{2}}$,
and the forgetful map $p:H^{1}(X,Z(\omega),\mathbb{R})\rightarrow
H^{1}(X,\mathbb{R})$ from relative to absolute homology. Denote by
$||\cdot||_{H}$ the norm induced by this inner product on
$H^{1}(X,\mathbb{R})$ and use the same notation to define a norm on
$H^{1}(X,Z(\omega),\mathbb{R})$ via

$||v||_{H}=||p(v)||_{H}+\sum\limits_{(p,p^{\prime})\in Z(\omega)\times
Z(\omega)}\bigg{\lvert}\int_{\gamma_{p,p^{\prime}}}(v-\mathbf{h})\bigg{\rvert}$

where $\mathbf{h}$ is the harmonic 1-form representing $p(v)$ and
$\gamma_{p,p^{\prime}}$ is any path connecting $p$ to $p^{\prime}$. The norm
is independent of the choice of path. We will denote the Hodge norm distance
by $d_{H}(\cdot,\cdot)$.

######  Corollary 4.4.1.

For $\mu_{MV}$-a.e. $x=(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$ and the
decomposition of the tangent bundle
$T_{x}\Omega\mathcal{M}_{g}(\sigma)\cong\mathbf{v}(x)\oplus E^{u}(x)\oplus
E^{s}(x)$, we have the following. There is an $a_{t}$-invariant function
$\lambda:\Omega_{1}\mathcal{M}_{g}(\sigma)\rightarrow\mathbb{R}^{+}$ with
$\lambda(x)<1$, so that for every $e_{0}>0$, there is an $a_{t}$-invariant
function $E:\Omega_{1}\mathcal{M}_{g}(\sigma)\rightarrow(0,\epsilon_{0})$ and
a function $C(x)>0$ on $\Omega_{1}\mathcal{M}_{g}(\sigma)$ where

$||D_{a_{t}}(x)v||_{H}\leq C(x)(\lambda(x))^{t}||v||_{H}$

for any $v\in E^{s}(x)$, and $D$ is the derivative. Further, $C(a_{t}x)\leq
C(x)e^{E(x)t}$.

###  4.3 Norm comparisons and closing lemma

It will be helpful for our purposes to know that the Teichmüller norm
$||\cdot||_{\mathcal{T}}$ defined in subsection 2.2 is commensurable with the
AGY-norm $||\cdot||$ defined in Definition 2.4 and the Hodge norm defined in
Section 4.2, on compact sets. We would like a quantitative comparison of these
norms, with explicit dependence on the compact set. For this, we will use the
main results in [45] due to Su and Zhang and the main results in [24] due to
Kahn and Wright.

For a quadratic differential $q$ on a Riemann surface $X$, there exists a
canonical double cover $\hat{X}$ and an Abelian differential $\omega$ on
$\hat{X}$ so that the pullback of $q$ to $\hat{X}$ is $\omega^{2}$. The set of
regular points includes the even zeroes of $q$, but the cover has ramification
at the odd zeroes and poles. An even zero of $q$ of order $n_{i}$ corresponds
to two zeroes of $\omega$ of orders $n_{i}/2$, and an odd zero of $q$ of order
$n_{i}$ corresponds to a zero of order $n_{i}+1$. The Deck group of $\hat{X}$
is an involution $\tau$; a conformal automorphism of $\hat{X}$ such that
$\tau^{*}(\omega)=-\omega$. For $g\geq 2$, we will denote by
$\mathcal{Q}_{1}\mathcal{M}_{g}(1,...,1)$ the principal stratum of unit-area
quadratic differentials; that is, the space of quadratic differentials $(X,q)$
with $4g-4$ simple zeroes. The tangent space of
$\mathcal{Q}_{1}\mathcal{M}_{g}(1,...,1)$ at $(X,q)$ can be identified with
$H^{1}_{-1}(\hat{X},Z(q),\mathbb{C})$, the $-1$ eigenspace for the involution
action on $H^{1}(\hat{X},Z(q),\mathbb{C})$, where $Z(q)$ denotes the zeros of
$q$. When $q$ has only odd zeroes, $H^{1}_{-1}(\hat{X},Z(q),\mathbb{C})$ can
be identified with $H^{1}_{-1}(\hat{X},\mathbb{C})$. Observe that in this
context, $\mathrm{inj}(X,q)$ is still well-defined, as is the norm in
Definition 2.4. Let
$\psi:\mathcal{Q}_{1}\mathcal{M}_{g}(1,...,1)\rightarrow\mathcal{M}_{g}$ be
the natural map $(X,q)\mapsto X$, and denote by $D$ its derivative.

######  Theorem 4.5 (Kahn-Wright, [24]).

Let $(X,q)\in\mathcal{Q}_{1}\mathcal{M}_{g}(1,...,1)$, and
$P:\hat{X}\rightarrow X$ such that $P^{*}q=\omega^{2}$. Then, for any $v\in
H^{1}_{-1}(\hat{X},\mathbb{C})$, we have

$||v||_{H}\leq||D\psi(v)||_{\mathcal{T}}\leq\frac{4}{\sqrt{\mathrm{inj}(X,q)}}||v||_{H}.$
(4.2)

######  Theorem 4.6 (Su-Zhang,[45]).

Let $(X,q)\in\mathcal{Q}_{1}\mathcal{M}_{g}(1,...,1)$, and
$P:\hat{X}\rightarrow X$ such that $P^{*}q=\omega^{2}$. Then, for any $v\in
H^{1}_{-1}(\hat{X},\mathbb{C})$, we have

$\frac{\mathrm{inj}({X},q)}{4\sqrt{2}}||v||\leq||D\psi(v)||_{\mathcal{T}}\leq\frac{16}{\sqrt{\mathrm{inj}(X,q)\pi}}||v||.$
(4.3)

We remark that every $(X,\omega)\in\Omega_{1}\mathcal{M}_{2}(2)$ is the
holonomy double cover of a unique surface in the principal stratum (with
possibly simple poles) $\mathcal{Q}_{1}\mathcal{M}_{g}(1,-1,-1,-1,-1,-1)$, and
Theorems 4.5 and 4.6 apply in this case. See Subsection 3.3 in [24] for an
application of this fact to the sharpness of the bounds in Theorem 4.5. Denote
the Teichmüller distance by $d_{\mathcal{T}}(\cdot,\cdot)$. With this notion
of distance, we require a closing lemma for the hyperbolic action of $a_{t}$
on a stratum. This statement weakly generalizes the usual Anosov closing lemma
for the geodesic flow $a_{t}$ on the unit tangent bundle of a hyperbolic
surface $T^{1}M=\mathrm{PSL}(2,\mathbb{R})/\Gamma$. Roughly, the Anosov
closing lemma in this context states that if for $x\in T^{1}M$ and $t>0$,
$d(x,a_{t}(x))<\epsilon$, then there exists an $x_{0}\in T^{1}M$ lying on a
periodic orbit, such that $d(x,x_{0})\ll\epsilon$. Since the geodesic flow on
a stratum isn’t uniformly hyperbolic, but only uniformly hyperbolic in compact
sets in the sense of Corollary 4.4.1, this requires a slight modification.
This following is a consequence of the arguments due to Eskin-Mirzakhani-Rafi
[14], Lemma 8.1. and to Hamenstädt [21]

######  Theorem 4.7 ([14],Lemma 8.1.,[21]).

Let $\mathcal{K}\subset\Omega_{1}\mathcal{M}_{g}(\sigma)$ be a compact set.
Given $(X,\omega)\in\mathcal{K}$ and $\delta>0$, there exists $t_{0}>0$ so
that the following holds. Suppose that $a_{t}(X,\omega)$ stays in
$\mathcal{K}$ for all $t\geq 0$ and $d_{H}((X,\omega),a_{t}(X,\omega))<\delta$
for $t>t_{0}$. Then, there exists nearby $(X^{\prime},\omega^{\prime})$ lying
on a closed Teichmüller geodesic of length within $\delta$ of $t$, and
$d_{\mathcal{T}}((X,\omega),(X^{\prime},\omega^{\prime}))<\delta$.

The proof relies on following the argument of the usual Anosov closing lemma
for uniformly hyperbolic flows, which itself relies on the contraction mapping
principle. Note that by Corollary 4.4.1, if one chooses
$x\in\mathcal{K}=\Omega_{1}\mathcal{M}_{\epsilon}$, $\epsilon_{0}$ small
enough, and $\lambda(x)=\lambda(a_{t}x)=e^{\kappa t}$, we may set
$\epsilon_{0}+\kappa=\epsilon_{x}+\kappa^{\prime}$ where $\epsilon_{x}$ and
$\kappa^{\prime}$ are both negative. Using the fact that $C(a_{t}x)\leq
C(x)e^{\epsilon_{0}t}$ we have

$||D_{a_{t}}(x)v||_{H}\leq e^{\kappa^{\prime}t}||v||_{H}$

if $C(x)e^{\epsilon_{x}t}<1$, i.e.,
$t>\mathrm{log}(1/C(x))\frac{1}{\epsilon_{x}}$, which is hyperbolicity with
hyperbolic factor $e^{\kappa^{\prime}t}$. Observe that $C(x)$ and the size of
an $\epsilon_{x}$ one can take depend only on the injectivity radius of $x$.
We also remark that Theorem 4.7 is stated for the geodesic flow $a_{t}$, but
is valid for any one-parameter hyperbolic action. The following theorem of
McMullen is crucial to our method, and is true only in stratum
$\Omega_{1}\mathcal{M}_{2}(2)$.

######  Theorem 4.8 ([32], Theorem 5.8).

Let $(X,\omega)\in\Omega_{1}\mathcal{M}_{2}(2)$ and suppose
$\gamma\in\mathrm{SL}(X,\omega)$ is a hyperbolic element of
$\mathrm{SL}(2,\mathbb{R})$. Then, $(X,\omega)$ lies on a Teichmüller curve.

###  4.4 Finding elements of the Veech group

Henceforth, for $v\in H^{1}(X,Z(\omega),\mathbb{C})$, we will suppress the
reference to the tangent space point, and simply write $||v||$. Recall that we
have a decomposition

$T_{\mathbb{C}}(\Omega_{1}\mathcal{M}_{g}(2))=T_{\mathbb{C}}^{st}(\Omega_{1}\mathcal{M}_{g}(2))\oplus
T_{\mathbb{C}}^{bal}(\Omega_{1}\mathcal{M}_{g}(2)).$

We also remind the reader that for $e^{-0.01t}<\beta<1$ and every $t\geq 0$,
we consider the "smeared" geodesic push

$E_{t}=B_{\beta}\cdot
a_{t}\cdot\left\\{u_{s}:s\in[0,1]\right\\}\subset\mathrm{SL}(2,\mathbb{R})$

where
$B_{\beta}:=\left\\{u_{s}^{\intercal}:|s|\leq\beta\right\\}\cdot\left\\{a_{t}:|t|\leq\beta\right\\}$
and $u_{s}^{\intercal}$ is the transpose of $u_{s}$. For any such $E_{t}$ as
above, and any $(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(2)$, the map
$E_{t}\rightarrow\Omega_{1}\mathcal{M}_{g}(2)$ given by $h\mapsto
h\cdot(X,\omega)$ is well-defined, since any $h\in E_{t}$ is in particular an
element of $\mathrm{SL}(2,\mathbb{R})$. It is is injective when, for any lift
of $(X,\omega)$, $(\tilde{X},\tilde{\omega})$, the restricted projection map
$\pi|_{B_{r}(\tilde{X},\tilde{\omega})\cap
E_{t}\cdot(\tilde{X},\tilde{\omega})}:\Omega_{1}\mathcal{T}_{g}(2)\rightarrow\Omega_{1}\mathcal{M}_{g}(2)$
is injective for any $r>0$. To make precise the definition of a "small
dimension transverse direction", we need a notion of transverse energy and a
Margulis function. Let $t\geq 0$, and
$(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(2)$, and assume that the map
$E_{t}\rightarrow\Omega_{1}\mathcal{M}_{g}(2)$ described above is injective.
For every $z\in E_{t}\cdot(X,\omega)$, set

$I_{bal,t}(z):=\left\\{w\in T_{z}^{bal}:0<||w||<r(z)\ \textrm{and}\
\mathrm{exp}_{z}(w)\in E_{t}\cdot(X,\omega)\right\\}$

where $T_{z}^{bal}$ is the balanced tangent space at $z$, $\mathrm{exp}$ is
the exponential map in Lemma 2.6 and $0<r(z)<1/C_{1}$ is defined in the
following way. Suppose $\tilde{r}(z)>0$ is the maximum $r$ so that the
restriction of the projection map from the Teichmüller space to moduli space
$\pi|_{B_{r}(\tilde{X},\tilde{\omega})}:\Omega_{1}\mathcal{T}_{g}(2)\rightarrow\Omega_{1}\mathcal{M}_{g}(2)$
is injective. Let $0<\theta<0.01$ and
$B^{\mathrm{SL}(2,\mathbb{R})}_{\theta}:=\left\\{u_{s}^{\intercal}:|s|\leq\theta\right\\}\cdot\left\\{a_{t}:|t|\leq\theta\right\\}\cdot\left\\{u_{s}:|s|\leq\theta\right\\}$.
Assume $\theta(z)$ is the supremum over all $\theta$ so that the following
holds; if $z^{\prime}=h\cdot\mathrm{exp}_{z}(w)$, and $h\in
B^{\mathrm{SL}(2,\mathbb{R})}_{\theta}$ is a non-identity element, there
exists no $w^{\prime}\in T_{z}^{bal}$ such that
$\mathrm{exp}_{z}(w^{\prime})=z^{\prime}$. Define

$r(z):=\textrm{min}\left\\{\theta(z),\tilde{r}(z),\mathrm{inj}(z)\right\\}$

where $\mathrm{inj}(z)$ is defined as in Section 3. Note that for a fixed $z$,
$I_{bal}(z)$ is a finite subset of $T_{\omega}^{bal}$, by definition and
because $E_{t}$ is bounded. Let $0<\nu<1$, and define the following Margulis
function $f_{t}:E_{t}\cdot(X,\omega)\rightarrow(0,\infty)$ by

$f_{t}(z)=\begin{cases}\sum_{0\neq w\in I_{bal}(z)}||w||^{-\nu}&\text{if}\
I_{bal}(z)\neq 0\\\ {r(z)}^{-\nu}&otherwise.\par\end{cases}$ (4.4)

We would like a quantitative bound on the number of elements of $I_{bal,t}(z)$
in terms of $t$. It is more delicate to obtain such a bound in our setting,
due to the inhomogeneous nature of moduli space. We require the following
Lemma due to Avila-Gouëzel-Yoccoz.

######  Lemma 4.9 (Lemma 5.2, [4]).

Let $x=(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$. For $v\in
H^{1}(X,Z(\omega),\mathbb{C})$ a tangent vector, $t\geq 0$, and $s\in[0,1]$,
we have the following inequalities

$e^{-2-2t}||v||\leq||(a_{t}u_{s})_{*}v||\leq e^{2+2t}||v||$ (4.5)

where $(a_{t}u_{s})_{*}v$ is the pushforward of the vector $v$ under the
$a_{t}u_{s}$ action.

######  Lemma 4.10.

Let $z\in E_{t}\cdot(X,\omega)\in\Omega_{1}\mathcal{M}_{2}(2)$, and assume
$(X,\omega)\in\Omega_{1}\mathcal{M}_{c}$. That is, that
$\mathrm{inj}(X,\omega)\geq c$. Then, $\\#I_{bal,t}(z)\ll e^{6\kappa_{4}t}$
for some $\kappa_{4}$ depending only on the stratum.

###### Proof.

We first show that, for any $h\in E_{t}$, $r(hz)\gg e^{-\kappa_{4}t}r(z)$ for
some $\kappa_{4}$ depending only on our stratum
$\Omega_{1}\mathcal{M}_{2}(2)$. Note, first, that since the $a_{t}$ action can
only shorten a saddle connection by at most a factor of $e^{-t}$,
$\mathrm{inj}(hz)\geq e^{-t}\mathrm{inj}(z)$, and so certainly
$\mathrm{inj}(hz)\gg e^{-\kappa_{4}t}\mathrm{inj}(z)$ for any $\kappa_{4}>1$.
For the case of $\tilde{r}(z)$, we have by Proposition 4.3 that
$\tilde{r}(hz)\gg e^{-C_{5}\kappa_{3}t}r(z)$, and consequently, treating the
case of $\theta(hz)$ suffices.

We now show that $\theta(hz)\gg e^{-4t}\theta(z)$. Specifically, we will show
that $\theta(hz)\geq\sqrt{2}/(4\sqrt{c\pi})e^{-2-2t}\theta(z)$. Unraveling the
definition of $\theta(z)$, we know that for any $\delta>0$ there exist
$w,w^{\prime}\in T^{bal}_{z}$ so that $\mathrm{exp}_{z}(w)\in
E_{t}\cdot(X,\omega)$ and $\mathrm{exp}_{z}(w^{\prime})\in
E_{t}\cdot(X,\omega)$, with
$\mathrm{exp}_{z}(w^{\prime})=h\cdot\mathrm{exp}_{z}(w)$ and $h\in
B^{\mathrm{SL}(2,\mathbb{R})}_{\theta(z)-\delta}$. Recall that the
decomposition of the tangent bundle into balanced and standard subbundles is
$\mathrm{SL}(2,\mathbb{R})$-invariant. Therefore, $h_{*}(w)$ and
$h_{*}(w^{\prime})\in T^{bal}_{hz}$. We may assume $\mathrm{exp}_{hz}(h_{*}w)$
and $\mathrm{exp}_{hz}(h_{*}w^{\prime})$ are well-defined inside a local
period coordinate chart; if they are not, then $\theta(hz)>\tilde{r}(hz)\gg
e^{-C_{5}\kappa_{3}t}\tilde{r}(z)>e^{-C_{5}\kappa_{3}t}r(z)$, and by the
definition of
$r(hz):=\mathrm{min}\left\\{\theta(hz),\tilde{r}(hz),\mathrm{inj}(hz)\right\\}$,
$r(hz)\gg e^{-C_{5}\kappa_{3}t}r(z)$. Taking $\kappa_{4}=\\\
\mathrm{max}\left\\{1,C_{5}\kappa_{3}\right\\}$, we are done.

Since $h_{*}(w)$ and $h_{*}(w^{\prime})\in T^{bal}_{hz}$, therefore are
transversal to the $\mathrm{SL}(2,\mathbb{R})$-orbit of $(X,\omega)$, we
deduce $\mathrm{exp}_{hz}(h_{*}w)$ and $\mathrm{exp}_{hz}(h_{*}w^{\prime})$
intersect the orbit $E_{t}\cdot(X,\omega)$. This is an affine subspace in this
local period coordinate chart. The affine geodesic (straight line in local
period coordinates) connecting $\mathrm{exp}_{hz}(h_{*}w)$ and
$\mathrm{exp}_{hz}(h_{*}w^{\prime})$ lies inside this affine subspace, and
altogether we have
$\mathrm{exp}_{hz}(h_{*}w)=h^{\prime}\cdot\mathrm{exp}_{hz}(h_{*}w^{\prime})$
with $h^{\prime}\in E_{t}$. We would like an upper bound on
$||h_{*}(w^{\prime})-h_{*}(w)||$ in terms of the quantity
$\theta(hz)(4/\pi\sqrt{2})$. By Equation 4.3, we have
$d_{\mathcal{T}}(\mathrm{exp}_{h\cdot z}(h_{*}w^{\prime}),\mathrm{exp}_{h\cdot
z}(h_{*}w))\geq(c/4\sqrt{2})||h_{*}(w^{\prime})-h_{*}(w)||$, and so

$\displaystyle||h_{*}(w^{\prime})-h_{*}(w)||\leq
e^{t}(4\sqrt{2}/c)\theta(hz).$ (4.6)

We now apply Lemma 4.9 to see that

$\displaystyle||h_{*}(w^{\prime})-h_{*}(w)||$
$\displaystyle=||h_{*}(w^{\prime}-w)||$ $\displaystyle\geq
e^{-2-2t}||w^{\prime}-w||.$ (4.7)

We combine Equations 4.6 and 4.4, and a similar estimate as Equation 4.6 for
$||w^{\prime}-w||$ to obtain $\theta(hz)\geq
e^{-2-4t}(4\sqrt{2}/c)(\sqrt{c\pi}/16)(\theta(z)-\delta))$. Choosing $\delta$
to be arbitrarily close to $0$, we obtain

$\displaystyle\theta(hz)\geq\sqrt{2}/(4\sqrt{c\pi})e^{-2-4t}\theta(z).$ (4.8)

Taking $\kappa_{4}=\mathrm{max}\left\\{C_{5}\kappa_{3},4\right\\}$, we are
done. Denote by $m_{\mathrm{SL}(2,\mathbb{R})}$ the Haar measure on
$\mathrm{SL}(2,\mathbb{R})$. We observe that for any $z\in
E_{t}\cdot(X,\omega)$ and $w\in I_{bal,t}(z)$, we have

$B^{\mathrm{SL}(2,\mathbb{R})}_{\sqrt{2c}/(4\sqrt{\pi})e^{-2-\kappa_{4}t}}\mathrm{exp}_{z}(w)\subset
E_{t+}\cdot(X,\omega)$

where
$E_{t+}:=B^{\mathrm{SL}(2,\mathbb{R})}_{3\sqrt{2c}/(4\sqrt{\pi})e^{-2-\kappa_{4}t}}\cdot
E_{t}$. We now have, by the definition of $I_{bal,t}(z)$, that for distinct
$w,w^{\prime}\in I_{bal,t}(z)$

$B^{\mathrm{SL}(2,\mathbb{R})}_{\sqrt{2c}/(4\sqrt{\pi})e^{-2-\kappa_{4}t}}\mathrm{exp}_{z}(w)\cap
B^{\mathrm{SL}(2,\mathbb{R})}_{\sqrt{2c}/(4\sqrt{\pi})e^{-2-\kappa_{4}t}}\mathrm{exp}_{z}(w^{\prime})=\emptyset.$
(4.9)

Since $m_{\mathrm{SL}(2,\mathbb{R})}(E_{t+})\ll e^{2t}$ and
$m_{\mathrm{SL}(2,\mathbb{R})}(B^{\mathrm{SL}(2,\mathbb{R})}_{ce^{-2-\kappa_{4}t}\pi\sqrt{2}/4})\gg
e^{-3\kappa_{4}t}$, the claim is proven.

∎

The following elementary observation will be convenient for us.

######  Proposition 4.11.

Let $c\in\mathbb{R},D>0,\eta>0$. Assume
$\mathrm{max}\left\\{a_{i}\right\\}\gg\eta^{2}$. The set of all
$x\in\mathbb{R}$ such that

$|a_{1}e^{-Dt}+a_{2}(x-c)-a_{3}e^{Dt}(x-c)^{2}|\leq Ce^{(-D+1)t}$ (4.10)

has measure $\ll\eta^{-4}e^{(\frac{-D+1}{2})t}$.

###### Proof of Theorem 1.2.

In the proceeding discussion, we take $\beta=e^{-\frac{1}{6\kappa_{4}+100}t}$.
Lemma 3.3, guarantees the existence constants $\alpha,C_{3},C_{4}>0$ so that
for any $0<\epsilon,\beta<1$ and
$(X,\omega)\in\Omega_{1}\mathcal{M}_{g}(\sigma)$ the following holds. Let
$I\subset[-10,10]$ be an interval and assume $|I|\geq\beta$. Then,

${\big{|}\left\\{s\in
I:\mathrm{inj}\left(a_{t}u_{s}(X,\omega)\right)<{\epsilon}\right\\}\big{|}}<C_{4}{\epsilon}^{\alpha}|I|,$
(4.11)

so long as $t\geq\big{\lvert}\mathrm{log}\left(\beta\
\mathrm{inj}(X,\omega)\right)^{-1}\big{\rvert}+C_{3}$. We will assume
$t\geq\mathrm{max}\left\\{\big{\lvert}\mathrm{log}\left(\beta\
\mathrm{inj}(X,\omega)\right)^{-1}\big{\rvert}+C_{3},\mathrm{log}(1/C(x))\frac{1}{\epsilon_{x}}\right\\}$
where $C(x)$ and $\epsilon_{x}$ depend only on $(X,\omega)$ as per the
discussion in Subsection 4.3. We fix such an interval $I$ that contains the
subinterval $[0,1]$ and assume $t$ satisfies the above lower bound for the
remainder of the proof. Let $s^{\prime}\in[0,1]$ be such that
$(X^{\prime},\omega^{\prime})=a_{t}u_{s^{\prime}}(X,\omega)\in\Omega_{1}\mathcal{M}_{\beta}$,
using the notation of Definition 3.1. We will show that if option $2$ in
Theorem 1.2 fails, then the following holds. For every
$(X^{\prime},\omega^{\prime})$ there exists an interval
$I(\omega^{\prime})\subset[0,1]$ with
$\big{|}[0,1]\big{\backslash}I(\omega^{\prime})\big{|}\leq
2C_{3}\beta^{\alpha}$ so that for all $s\in I(\omega^{\prime})$

  1. (a)

$a_{7t}u_{s}(X^{\prime},\omega^{\prime})\in\Omega_{1}\mathcal{M}_{\beta}$,

  2. (b)

$h\rightarrow a_{7t}u_{s}(X^{\prime},\omega^{\prime})$ is an injective map
$E_{t}\rightarrow\Omega_{1}\mathcal{M}_{2}(2)$,

  3. (c)

for all $z\in E_{t}\cdot a_{7t}u_{s}(X^{\prime},\omega^{\prime})$, we have
$f_{t}(z)\leq e^{(6\kappa_{4}+100)t}$.

Note that $u_{s}a_{t}=a_{t}u_{se^{-t}}$, so that

$\displaystyle a_{7t}u_{s}a_{t}u_{s^{\prime}}(X,\omega)$ $\displaystyle=$
$\displaystyle=a_{7t}a_{t}u_{se^{-t}}u_{s^{\prime}}(X,\omega)$
$\displaystyle=a_{8t}u_{s^{\prime}+se^{-t}}(X,\omega).$

which implies that option $1$ in Theorem 1.2 holds with $D=6\kappa_{4}+100$.
Assume to the contrary that there exists $(X^{\prime},\omega^{\prime})$ and a
subset $I^{\prime}_{bad}\subset[0,1]$ with
$\big{|}I^{\prime}_{bad}\big{|}>2C_{3}\beta^{\alpha}$ so that one of
$(a),(b),(c)$ is false. By 4.11, ${\big{|}\left\\{s\in
I:\mathrm{inj}\left(a_{7t}u_{s}(X^{\prime},\omega^{\prime})\right)<{\beta}\right\\}\big{|}}<C_{3}{\beta}^{\alpha}$,
there exists $I_{bad}\subset[0,1]$ with
$\big{|}I_{bad}\big{|}>1-C_{3}\beta^{\alpha}>2C_{3}\beta^{\alpha}-C_{3}\beta^{\alpha}=C_{3}\beta^{\alpha}$
so that for all $s\in I_{bad}$,
$a_{7t}u_{s}(X^{\prime},\omega^{\prime})\in\Omega_{1}\mathcal{M}_{\beta}$ but
one of $(b)$ and $(c)$ is false. We will show that this implies option $2$ of
Theorem 1.2.

We will prove the statement for option $(c)$ as option $(b)$ is similar. By
definition of $I_{bad}$, we have that for any $s\in I_{bad}$, there exists a
$z\in E_{t}\cdot a_{7t}u_{s}(X^{\prime},\omega^{\prime})$ so that
$f_{t}(z)>e^{(6\kappa_{4}+100)t}$. Since
$a_{7t}u_{s}(X^{\prime},\omega^{\prime})\in\Omega_{1}\mathcal{M}_{\beta}$, we
have, by the proof of Lemma 4.10,

$r(ha_{7t}u_{s}(X^{\prime},\omega^{\prime})\gg\beta e^{-2t}$

for all $h\in E_{t}$. By the definition of $f_{t}$, if
$I_{bal,t}(z)=\left\\{0\right\\}$, then $f_{t}(z)\ll\beta^{-1}e^{2t}$. Since
we have assumed $t\geq\big{\lvert}\mathrm{log}\left(\beta\
\mathrm{inj}(X,\omega)\right)^{-1}\big{\rvert}+C_{3}$, we have

$\displaystyle f_{t}(z)$ $\displaystyle\ll\beta^{-1}e^{2t}$
$\displaystyle\leq\beta^{-5}e^{2C_{3}}\ll\beta^{-5}.$

If we take $t$ large enough, this contradicts
$f_{t}(z)>e^{(6\kappa_{4}+100)t}$, so we may assume
$I_{bal,t}(z)\neq\left\\{0\right\\}$. By Lemma 4.10, $\\#I_{bal,t}(z)\ll
e^{6\kappa_{4}t}$. By the definition of $I_{bal,t}(z)$, there exists $w\in
T_{z}^{bal}$ with $||w||\ll e^{-100t}$.
Altogether, there exists some $\bar{z}\in E_{t}\cdot
a_{7t}u_{s}(X^{\prime},\omega^{\prime})$ so that
$\bar{z}=\mathrm{exp}_{z}(\omega)$ with $||w||\ll e^{-100t}$, and for some
$\mathbf{h}_{2}\in E_{t}$ we have
$\bar{z}=\mathbf{h}_{2}a_{7t}u_{s}(X^{\prime},\omega^{\prime})$. Let
$\mathbf{h}_{1}\in E_{t}$ be such that
$z=\mathbf{h}_{1}a_{7t}u_{s}(X^{\prime},\omega^{\prime})$ and note that
$\bar{z}=\mathbf{h}_{2}\mathbf{h}^{-1}_{1}z$. Henceforth, for such
$\mathbf{h}_{2}$, $\mathbf{h}_{2}$, we will use the shorthand
$\mathbf{h}_{s}:=\mathbf{h}_{2}\mathbf{h}^{-1}_{1}$.

Distinct conjugacy classes of mapping class group elements

The structure of the remainder of the proof is an adaptation of the proof of
Proposition 6.1 in [26] to moduli space. The proof is quite similar, though
requires modifications significant enough to merit a detailed treatment here.

Since $|I_{bad}|>C_{3}\beta^{\alpha}$, one can find intervals
$J,J^{\prime}\subset[0,1]$ with $d(J,J^{\prime})\gg\beta^{\alpha}$,
$|J|,|J^{\prime}|\gg\beta^{\alpha}$ and $|J\cap I_{bad}|\geq\beta$,
$|J^{\prime}\cap I_{bad}|\geq\beta$. Set $J_{\beta}=J\cap I_{bad}$. We will
further use the shorthand $\mathbf{p}_{s}:=a_{7t}u_{s}$. Note that we have the
following inequalities.

$\displaystyle
d_{\mathcal{T}}\left((X^{\prime},\omega^{\prime}),(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}(X^{\prime},\omega^{\prime})\right)$
$\displaystyle\ll
e^{14t}d_{\mathcal{T}}\left(\mathbf{p}_{s}(X^{\prime},\omega^{\prime}),\mathbf{h}_{s}\mathbf{p}_{s}(X^{\prime},\omega^{\prime})\right)$
$\displaystyle\ll e^{-84t}.$

Thus, there exist lifts $(\tilde{X}^{\prime},\tilde{\omega}^{\prime})$ of
$(X^{\prime},\omega^{\prime})$ and, for each lift, an element
${\mathbf{\gamma}}_{s}$ in the mapping class group so that

$d_{\mathcal{T}}\left((\tilde{X}^{\prime},\tilde{\omega}^{\prime}),(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}{\mathbf{\gamma}}_{s}(\tilde{X}^{\prime},\tilde{\omega}^{\prime})\right)\ll
e^{-84t}$ (4.12)

using the fact the mapping class group commutes with
$\mathrm{SL}(2,\mathbb{R})$. Suppose that
$(\tilde{\tilde{X}}^{\prime},\tilde{\tilde{\omega}}^{\prime})$ is a different
lift of $(X,\omega)$,
$\gamma\cdot(\tilde{\tilde{X}}^{\prime},\tilde{\tilde{\omega}}^{\prime})=(\tilde{X}^{\prime},\tilde{\omega}^{\prime})$,
for some $\gamma$. Then,

$d_{\mathcal{T}}\left((\tilde{\tilde{X}}^{\prime},\tilde{\tilde{\omega}}^{\prime}),(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}\gamma{\mathbf{\gamma}}_{s}(\gamma)^{-1}(\tilde{X}^{\prime},\tilde{\omega}^{\prime})\right)=d_{\mathcal{T}}\left((\tilde{X}^{\prime},\tilde{\omega}^{\prime}),(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}{\mathbf{\gamma}}_{s}(\tilde{X}^{\prime},\tilde{\omega}^{\prime})\right).$

So, to each $s$, we associate a conjugacy class $[\mathbf{\gamma}_{s}]$.
Suppose that for $s\neq s^{\prime}$,
$[\mathbf{\gamma}_{s}]=[\mathbf{\gamma}_{s^{\prime}}]$. Then, for a lift
$(\tilde{X}^{\prime},\tilde{\omega}^{\prime})$, we have

$d_{\mathcal{T}}\left((\tilde{X}^{\prime},\tilde{\omega}^{\prime}),(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}{\mathbf{\gamma}}_{s}(\tilde{X}^{\prime},\tilde{\omega}^{\prime})\right)=d_{\mathcal{T}}\left((\tilde{X}^{\prime},\tilde{\omega}^{\prime}),(\mathbf{p}_{s^{\prime}})^{-1}\mathbf{h}_{s^{\prime}}\mathbf{p}_{s^{\prime}}{\mathbf{\gamma}}_{s}(\tilde{X}^{\prime},\tilde{\omega}^{\prime})\right).$
(4.13)

By the triangle inequality, and Equation 4.12 we have

$d_{\mathcal{T}}\left((\mathbf{p}_{s^{\prime}})^{-1}\mathbf{h}_{s^{\prime}}\mathbf{p}_{s^{\prime}}{\mathbf{\gamma}}_{s}(\tilde{X}^{\prime},\tilde{\omega}^{\prime}),(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}{\mathbf{\gamma}}_{s}(\tilde{X}^{\prime},\tilde{\omega}^{\prime})\right)\ll
e^{-84t}.$ (4.14)

By the definition of Teichmüller distance, and using the first order
approximation $\mathrm{log}(x)\sim(x-1)$ for $x$ near one, we obtain from
Equation 4.14

$||(\mathbf{p}_{s^{\prime}})^{-1}\mathbf{h}_{s^{\prime}}\mathbf{p}_{s^{\prime}}(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}||\ll
e^{-68t}.$ (4.15)

By repeated applications of the facts on the matrix norm in Subsection 2.1, we
have

$\frac{||\mathbf{p}_{s^{\prime}}(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}(\mathbf{p}_{s^{\prime}})^{-1}||}{||\mathbf{h}_{s^{\prime}}||}\ll
e^{14t}||(\mathbf{p}_{s^{\prime}})^{-1}\mathbf{h}_{s^{\prime}}\mathbf{p}_{s^{\prime}}(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}||.$
(4.16)

By proper discontinuity of the action of the mapping class group, we may find
$\chi>0$ be such that for any non-identity (conjugacy class) $\gamma_{s}$ we
have for $\mathbf{h}_{s}=\begin{bmatrix}a_{1,s}&a_{2,s}\\\
a_{3,s}&a_{4,s}\end{bmatrix}$, where $|a_{i,s}|\leq 10e^{2t}$,

$||(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}(\mathbf{p}_{s})-I||=\Bigg{\lvert}\Bigg{\lvert}u_{-s}\begin{bmatrix}a_{1,s}&e^{-7t}a_{2,s}\\\
e^{7t}a_{3,s}&a_{4,s}\end{bmatrix}u_{s}-I\Bigg{\rvert}\Bigg{\rvert}\geq
20\beta^{2\alpha}\chi.$ (4.17)

From this, we conclude
$\mathrm{max}\left\\{e^{7t}|a_{3,s}|,|a_{1}-1|,|a_{4}-1|\right\\}\gg\beta^{2\alpha}$.
Set $\tau=e^{7t}(s^{\prime}-s)$. We compute

$\displaystyle\begin{split}u_{\tau}\mathbf{h}_{s}u_{-\tau}&=\mathbf{p}_{s^{\prime}}(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}(\mathbf{p}_{s^{\prime}})^{-1}\\\
&=\begin{bmatrix}a_{1,s}+a_{3,s}\tau&a_{2,s}+(a_{4,s}-a_{1,s})\tau-
a_{3,s}\tau^{2}\\\ a_{3,s}&a_{4,s}-a_{3,s}\tau\end{bmatrix}.\end{split}$
(4.18)

By Proposition 4.11, and the fact that
$\mathrm{max}\left\\{e^{7t}|a_{3,s}|,|a_{1}-1|,|a_{4}-1|\right\\}\gg\beta^{\alpha}$,
we may conclude that for every $s\in J_{\beta}$, the set of $s^{\prime}\in
J_{\beta}$ so that

$|a_{2,s}e^{-7t}+(a_{4}-a_{1})(s^{\prime}-s)-a_{3}e^{7t}(s^{\prime}-s)^{2}|\leq
e^{-6t}$ (4.19)

is of measure $\ll\beta^{-2\alpha}e^{-3t}$. If we denote by $J_{\beta,s}$ the
set of $s^{\prime}\in J_{\beta}$ for which Equation 4.19 is valid, and we let
$s^{\prime}\in J_{\beta}\setminus J_{\beta,s}$ then for all such $s^{\prime}$
we compute

$\frac{||u_{\tau}\mathbf{h}_{s}u_{-\tau}||}{||\mathbf{h}_{s^{\prime}}||}\gg
e^{-2t}$

using Equation 4.18 and the fact that $||\mathbf{h}_{s^{\prime}}||\ll e^{2t}$.
This is in contradiction to Equations 4.15 and 4.16. We conclude that for
$s\in J_{\beta}$, the set of $s^{\prime}\in J_{\beta}$ for which
$[\gamma_{s}]=[\gamma_{s^{\prime}}]$ has measure $\ll\beta^{-2\alpha}e^{-3t}$.
Therefore, the set of $s^{\prime}\in J_{\beta}$ for which
$(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}=(\mathbf{p}_{s^{\prime}})^{-1}\mathbf{h}_{s^{\prime}}\mathbf{p}_{s^{\prime}}$
is of measure $\ll\beta^{-2\alpha}e^{-3t}$. Since
$||(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}||\ll e^{9t}$, the set

$\left\\{(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s},\ s\in
J_{\beta}\right\\}$

has cardinality $\gg e^{2.9t}$.

Existence of hyperbolic element

We would like to show the group $\mathbf{G}$ generated by
$\langle(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s},\ s\in
I_{bad}\rangle$ cannot be unipotent; proceeding by contradiction, assume that

$(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}=u_{-s}\begin{bmatrix}a_{1,s}&e^{-7t}a_{2,s}\\\
e^{7t}a_{3,s}&a_{4,s}\end{bmatrix}u_{s}\in gUg^{-1}$ (4.20)

for all $s\in I_{bad}$, where $U=\left\\{\begin{bmatrix}1&r\\\
0&1\end{bmatrix},\,r\in\mathbb{R}\right\\}$ and
$g\in\mathrm{SL}(2,\mathbb{R})$. Since $\mathbf{G}$ is unipotent, we know

$\big{\lvert}\left\\{\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}:\
||\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}||\leq
e^{4t/3}\right\\}\big{\rvert}\ll e^{8t/3}.$

We have just shown
$\big{\lvert}\left\\{(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s},\ s\in
J_{\beta}\right\\}\big{\rvert}\gg e^{2.9t}$ so we can therefore conclude

$\big{\lvert}\left\\{\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}:\
||\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}||\geq e^{4t/3}\
\textrm{and}\ s\in J_{\beta}\right\\}\big{\rvert}\gg e^{t/4}.$ (4.21)

Let $\beta^{2\alpha}\chi\leq\Psi\leq e^{4t/3}$. A computation shows that if

$||(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}(\mathbf{p}_{s})-I||=\Bigg{\lvert}\Bigg{\lvert}u_{-s}\begin{bmatrix}a_{1,s}&e^{-7t}a_{2,s}\\\
e^{7t}a_{3,s}&a_{4,s}\end{bmatrix}u_{s}-I\Bigg{\rvert}\Bigg{\rvert}\geq
20\Psi$

then, for $t$ large enough

$|a_{3,s}|\geq\Psi e^{-7t}.$ (4.22)

By Equation 4.20, for each $s\in I_{bad}$, there is $r_{0}\in\mathbb{R}$ so
that

$u_{-s}\begin{bmatrix}a_{1,s}&e^{-7t}a_{2,s}\\\
e^{7t}a_{3,s}&a_{4,s}\end{bmatrix}u_{s}=g\begin{bmatrix}1&r_{0}\\\
0&1\end{bmatrix}g^{-1}.$ (4.23)

Let $s_{0}\in J^{\prime}\cap I_{bad}$. Setting
$u_{s_{0}}g=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}$, Equation 4.23 gives

$\begin{bmatrix}a_{1,s_{0}}&e^{-7t}a_{2,s_{0}}\\\
e^{7t}a_{3,s_{0}}&a_{4,s_{0}}\end{bmatrix}=\begin{bmatrix}1-acr_{0}&a^{2}r_{0}\\\
-c^{2}r_{0}&1+acr_{0}\end{bmatrix}$ (4.24)

for some $r_{0}\in\mathbb{R}$. Applying Equation 4.22 with
$\Psi=\beta^{2\alpha}\chi$, we have $|a_{3,s_{0}}|\geq
e^{-7t}\beta^{2\alpha}\chi$, and, since $g$ is fixed so its entries are
bounded by absolute constants, we may compare matrix entries on the right and
left hand sides to conclude $|a|\ll e^{-2.9t}$ and $|c|\gg 1$. Now, let $s\in
J_{\beta}$. By Equation 4.21, we may assume
$||(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}||\geq e^{4t/3}$. Set
$s_{1}=s-s_{0}$, $a^{\prime}_{2,s}=e^{-7t}a_{2,s}$,
$a^{\prime}_{3,s}=e^{7t}a_{3,s}$. By Equation 4.22 applied with
$\Psi=e^{4t/3}$, we deduce $|a^{\prime}_{3,s}|\geq e^{4t/3}$. For the sake of
once again comparing matrix entries, apply Equation 4.20 to obtain

$\displaystyle u_{-s_{1}}\begin{bmatrix}a_{1,s}&a^{\prime}_{2,s}\\\
a^{\prime}_{3,s}&a_{4,s}\end{bmatrix}u_{s_{1}}$
$\displaystyle=\begin{bmatrix}a_{1,s}-s_{1}a^{\prime}_{3,s}&a^{\prime}_{2,s}+(a_{4,s}-a_{1,s})s_{1}-a^{\prime}_{3,s}(s_{1})^{2}\\\
a^{\prime}_{3,s}&a_{4,s}+s_{1}a^{\prime}_{3,s}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1-acr&a^{2}r\\\ -c^{2}r&1+acr\end{bmatrix}$

where $r\in\mathbb{R}$. Compare matrix entries using $|a^{\prime}_{3,s}|\geq
e^{4t/3}$ and, by $d(J,J^{\prime})\gg\beta^{\alpha}$, the inequality
$\beta\leq\beta^{\alpha}\leq s_{1}\leq 1$. We conclude $|a|^{2}$ and $|c|^{2}$
are comparable in size. This contradicts $|a|\ll e^{-2.9t}$ and $|c|\gg 1$ for
$t$ large.

In light of the discussion in Subsection 2.4, we may assume
$(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s}$ are not elliptic, and we
have just demonstrated
$\langle(\mathbf{p}_{s})^{-1}\mathbf{h}_{s}\mathbf{p}_{s},\ s\in
I_{bad}\rangle$ is not unipotent. It is, in particular not generated by a
single parabolic element. It must therefore contain two transverse parabolic
elements, and by the ping-pong lemma must contain a hyperbolic element.
Considering traces, there must be an $\mathbf{h}_{s}$ hyperbolic.

All in all, there exist $z,\bar{z}\in E_{t}\cdot
a_{7t}u_{s}\cdot(X^{\prime},\omega^{\prime})$, $\bar{z}=\mathbf{h}_{s}\cdot z$
where $d_{\mathcal{T}}(z,\bar{z})\ll e^{-80t}$. By Theorem 4.5,
$d_{H}(z,\bar{z})\ll e^{-60t}$. By Theorem 4.7, there exists a point
$z^{\prime}$ fixed by $\mathbf{h}_{s}$ with $d_{\mathcal{T}}(z^{\prime},z)\ll
e^{-60t}$. In particular, by Equation 4.9, and the fact that
$d_{\mathcal{T}}(z^{\prime},z)\ll e^{-60t}$, $\gamma_{s}$ corresponding to
$\mathbf{h}_{s}$ must be non-trivial. The translation length
$l(\mathbf{h}_{s})$ satisfies $e^{-t/2}\ll\mathrm{log}(l(\mathbf{h}_{s}))\ll
e^{20t}$. The diagonalizable element $\mathbf{h}_{s}$ is in the Veech group of
$z^{\prime}$ and by Theorem 4.8, $z^{\prime}$ lies on a Teichmüller curve.
Theorem 1.2 follows from this, taking $D\geq 6\kappa_{4}+100$. ∎

## References

  * [1] V. Arnold, M. Atiyah, P. Lax, and B. Mazur.  Mathematics: Frontiers and perspectives.  The American Mathematical Monthly, 108, 04 2001.
  * [2] Masayuki Asaoka and Kei Irie.  A ${C}^{\infty}$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces.  Geometric and Functional Analysis, 26, 12 2015.
  * [3] Jayadev S Athreya.  Quantitative recurrence and large deviations for Teichmüller geodesic flow.  Geometriae Dedicata, 119:121–140, 2006.
  * [4] A. Avila, S. Gouëzel, and J.C. Yoccoz.  Exponential mixing for the Teichmüller flow.  Publ.math.IHES, 104:143–211, 2006.
  * [5] Artur Avila and Sebastien Gouezel.  Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow.  Annals of Mathematics, 178:385–442, 2010.
  * [6] Jean Bourgain, Alex Furman, Elon Lindenstrauss, and Shahar Mozes.  Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus.  Journal of the American Mathematical Society, 24(1):231–280, 2011.
  * [7] Jon Chaika, Osama Khalil, and John Smillie.  On the space of ergodic measures for the horocycle flow on strata of abelian differentials.  arXiv preprint arXiv:2104.00554, 2023.
  * [8] Jon Chaika, Jonathan Smillie, and Barak Weiss.  Tremors and horocycle dynamics on the moduli space of translation surfaces.  arXiv preprint arXiv:2004.04027, 2022.
  * [9] Manfred Einsiedler, Gregory Margulis, and Akshay Venkatesh.  Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces.  Inventiones mathematicae, 177(1):137–212, 2009.
  * [10] A. V. Eskin and Howard A. Masur.  Asymptotic formulas on flat surfaces.  Ergodic Theory and Dynamical Systems, 21:443 – 478, 2001.
  * [11] Alex Eskin and Maryam Mirzakhani.  Invariant and stationary measures for the SL(2,$\mathbb{R}$) action on moduli space.  Publications mathématiques de l’IHÉS, 127(1):95–324, 2018.
  * [12] Alex Eskin, Maryam Mirzakhani, and Amir Mohammadi.  Isolation, equidistribution, and orbit closures for the SL(2, $\mathbb{R}$) action on moduli space.  Annals of Mathematics, pages 673–721, 2015.
  * [13] Alex Eskin, Maryam Mirzakhani, and Amir Mohammadi.  Effective counting of simple closed geodesics on hyperbolic surfaces.  Journal of the European Mathematical Society, 24, 10 2021.
  * [14] Alex Eskin, Maryam Mirzakhani, and Kasra Rafi.  Counting closed geodesics in strata.  Inventiones mathematicae, 215(2):535–607, 2019.
  * [15] Simion Filip.  Zero lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle.  Duke Mathematical Journal, 166:657–706, 2014.
  * [16] Simion Filip.  Splitting mixed Hodge structures over affine invariant manifolds.  Annals of Mathematics, 183(2):681–713, 2016.
  * [17] Giovanni Forni.  Deviation of ergodic averages for area-preserving flows on surfaces of higher genus.  Annals of Mathematics, 155(1):1–103, 2002.
  * [18] H. Furstenberg.  Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.  Math. Systems Theory 1, pages 1–49, 1967.
  * [19] Ben Green and Terence Tao.  The quantitative behaviour of polynomial orbits on nilmanifolds.  Annals of Mathematics, pages 465–540, 2012.
  * [20] Ursula Hamenstaedt.  Dynamics of the Teichmüller flow on compact invariant sets.  Journal of Modern Dynamics, 4, 05 2007.
  * [21] Ursula Hamenstaedt.  Bowen’s construction for the teichmüller flow.  Journal of Modern Dynamics, 7, 07 2010.
  * [22] Pascal Hubert and Thomas A. Schmidt.  Infinitely generated Veech groups.  Duke Mathematical Journal, 123(1):49 – 69, 2004.
  * [23] Michael Hutchings.  Elementary spectral invariants and quantitative closing lemmas for contact three-manifolds.  arXiv preprint arXiv:2208.01767, 2024.
  * [24] Jeremy Kahn and Alex Wright.  Hodge and teichmüller.  Journal of Modern Dynamics, 18:149, 01 2022.
  * [25] A. Katok, A.B. Katok, and B. Hasselblatt.  Introduction to the Modern Theory of Dynamical Systems.  Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1995.
  * [26] E. Lindenstrauss and A. Mohammadi.  Polynomial effective density in quotients of $\mathbb{H}^{3}$ and $\mathbb{H}^{2}\times\mathbb{H}^{2}$.  Inventiones mathematicae, 231, 09 2022.
  * [27] Elon Lindenstrauss and Gregory Margulis.  Effective estimates on indefinite ternary forms.  Israel Journal of Mathematics, 203(1):445–499, 2014.
  * [28] Elon Lindenstrauss, Amir Mohammadi, and Zhiren Wang.  Effective equidistribution for some one parameter unipotent flows.  arXiv preprint arXiv:2211.11099, 2022.
  * [29] Elon Lindenstrauss, Amir Mohammadi, and Zhiren Wang.  Polynomial effective equidistribution.  arXiv preprint arXiv:2202.11815, 2022.
  * [30] Elon Lindenstrauss, Amir Mohammadi, Zhiren Wang, and Lei Yang.  An effective version of the Oppenheim conjecture with a polynomial error rate.  arXiv preprint arXiv:2305.18271, 2023.
  * [31] Curtis McMullen.  Billiards and Teichmüller curves on Hilbert modular surfaces.  Journal of the American Mathematical Society, 16(4):857–885, 2003.
  * [32] Curtis McMullen.  Dynamics of $\mathrm{SL}(2,\mathbb{R})$ over moduli space in genus two.  Annals of mathematics, ISSN 0003-486X, Vol. 165, Nº 2, 2007, pags. 397-456, 165, 03 2007.
  * [33] Curtis T. McMullen.  Teichmüller geodesics of infinite complexity.  Acta Mathematica, 191(2):191 – 223, 2007.
  * [34] Yair Minsky and Barak Weiss.  Nondivergence of horocyclic flows on moduli space.  Journal fur die Reine und Angewandte Mathematik, (552):131–177, 2002.
  * [35] Charles C. Pugh.  An improved closing lemma and a general density theorem.  American Journal of Mathematics, 89(4):1010–1021, 1967.
  * [36] Marina Ratner.  Rigidity of time changes for horocycle flows.  Acta Mathematica, 156:1–32, 1986.
  * [37] Marina Ratner.  On measure rigidity of unipotent subgroups of semisimple groups.  Acta Mathematica, 165(3-4):229–309, 1990.
  * [38] Marina Ratner.  On Raghunathan’s measure conjecture.  Annals of Mathematics, 134(3):545–607, 1991.
  * [39] Marina Ratner.  Raghunathan’s topological conjecture and distributions of unipotent flows.  Duke Mathematical Journal, 63:235–280, 1991.
  * [40] Anthony Sanchez.  Effective equidistribution of large dimensional measures on affine invariant submanifolds.  arXiv preprint arXiv:2306.06740, 2023.
  * [41] John Smillie, Peter Smillie, Barak Weiss, and Florent Ygouf.  Horospherical dynamics in invariant subvarieties.  arXiv preprint arXiv:2303.07188, 2023.
  * [42] John Smillie and Barak Weiss.  Minimal sets for flows on moduli space.  Israel Journal of Mathematics, 142(1):249–260, 2004.
  * [43] Omri Nisan Solan.  Critical exponent gap and leafwise dimension.  arXiv preprint arXiv:2404.00700, 2024.
  * [44] Andreas Strömbergsson.  An effective Ratner equidistribution result for SL(2, $\mathbb{R}$)$\ltimes$ $\mathbb{R}^{2}$.  Duke Mathematical Journal, 164(5):843–902, 2015.
  * [45] Weixu Su and Shenxing Zhang.  A comparison between Avila-Gouëzel-Yoccoz norm and Teichmüller norm.  Topology and its Applications, 345:108839, 2024.
  * [46] Alex Wright.  The field of definition of affine invariant submanifolds of the moduli space of Abelian differentials.  Geometry & Topology, 18(3):1323–1341, 2014.
  * [47] Lei Yang.  Effective version of Ratner’s equidistribution theorem for $\mathrm{SL}(3,\mathbb{R})$.  arXiv preprint arXiv:2208.02525, 2023.

Department of Mathematics and Statistics, Binghamton University. Binghamton,
New York, United States
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