# Bounds for the random walk speed
in terms of the distance

Aitor Azemar<EMAIL_ADDRESS>Mathematics & Statistics, University
of Glasgow G128QQ, UK

###### Abstract

Consider a closed surface $S$ with negative Euler characteristic, and an
admissible probability measure on the fundamental group of $S$ with finite
first moment with respect to some hyperbolic metric on $S$. Corresponding to
each point in Teichmüller space there is an associated random walk on the
hyperbolic plane. Azemar–Gadre–Gouëzel–Haettel–Lessa–Uyanik prove that the
drift of this random walk is a proper function on Teichmüller space, and that
this drift grows at least linearly with respect to the distance. In this paper
we refine the result. On the one hand, by considering Jenkins-Strebel
directions we show that the linear lower bound is sharp. On the other hand, we
show that for Lebesgue typical geodesics, the drift grows exponentially. We
also exhibit geodesics for which the growth oscillates between almost linear
and exponential. Furthermore, we show that the drift is a quasiconvex function
up to a multiplicative constant.

##  1 Introduction

Let $S$ be a closed oriented surface with negative Euler characteristic and
let $p\in S$ be a basepoint. Furthermore, let $\Gamma=\pi_{1}(S,p)$ and let
$\mu$ be a probability measure on $\Gamma$ that is admissible, i.e., the
semigroup generated by the support of $\mu$ is equal to $\Gamma$. For each
hyperbolic metric $\rho$ on $S$ we can define the length of an element
$g\in\Gamma$ as the minimal $\rho$-length of the free loops within the
homotopy class of $g$, which we denote $|g|_{\rho}$. Assume then that $\mu$
has finite first moment with respect to $\rho$ (and hence, with respect to any
other hyperbolic metric $\rho^{\prime},$ see Section 2 for details). Consider
a random walk $Z^{\mu}_{n}=g_{1}\dotsm g_{n}$ where $g_{i}$ are i.i.d.
elements of $\Gamma$ with distribution $\mu$. The _drift (or speed)_ of the
random walk on $\Gamma$ driven by $\mu$ with respect to $\rho$ is defined as

$\operatorname{Drift}_{\mu}(\rho)\coloneqq\lim_{n\to\infty}\frac{\lvert
Z_{n}^{\mu}\rvert_{\rho}}{n}$

The limit above exists and has constant value almost surely. That is, the
drift is a well-defined property of the random walk, depending only on
$\Gamma$, $\mu$ and $\rho$.

Denote as $\mathcal{T}(S)$ the space of $S$. It has been proven by
Azemar–Gadre–Gouëzel–Haettel–Lessa–Uyanik [AGG+nt, Theorem B] that the
function $\operatorname{Drift}:\mathcal{T}(S)\to\mathbb{R}_{+}$ grows at least
linearly with respect to the distance. In this paper we give more details
about this relation.

There is a natural identification between the set $Q^{1}(o)$ of unit area
quadratic differentials $q$ under the conformal structure defined by a
basepoint $o\in\mathcal{T}(S)$ and the set of geodesic rays starting at $o$.
For a given $q\in Q^{1}(o)$ we shall denote by
${\mathcal{R}(q;\cdot)}:[0,\infty)\to\mathcal{T}(S)$ the associated geodesic.
We find that whenever the quadratic differential is Jenkins–Strebel the linear
lower bound of the growth is sharp. More precisely, we show the following.

######  Theorem 1.1.

Let $\mu$ be an admissible measure on $\pi_{1}(S,p)$ with finite first moment.
Furthermore, let $o\in\mathcal{T}(S)$ be a basepoint, and let $q\in Q^{1}(o)$
be Jenkins–Strebel. Then, there are constants $c(o),C(q,o)>0$ such that, along
the geodesic ray ${\mathcal{R}(q;\cdot)}$,

$c(o)t\leq\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})\leq C(q,o)t$

for all $t\geq 1$.

Furthermore, we find that this is not the typical behaviour. As long as the
measure on the set of directions satisfies a thickness property, we get that
along typical geodesics the drift grows exponentially. For example, we find
the following.

######  Theorem 1.2.

Let $\mu$ be an admissible measure on $\pi_{1}(S,p)$ with finite first moment.
Furthermore, let $o\in\mathcal{T}(S)$ be a basepoint, $\theta<1$ and $\lambda$
be the Lebesgue measure on $Q^{1}(o)$. Then, for $\lambda$-almost all
directions $q\in Q^{1}(o)$ we have

$e^{\theta
t}<\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})<e^{\frac{1}{\theta}t},$

for any $t>t(q)$, where $t(q)$ is some finite time depending on $q$.

See Theorem 4.7 for a more general version of the previous result, which also
applies to harmonic measures on $Q^{1}(o)$ arising from random walks on the
mapping class group with finite first moments. We also find that these two
behaviours are not the only ones, and in fact there are many intermediate
behaviours. Precisely, we find the following.

######  Theorem 1.3.

Let $o\in\mathcal{T}(S)$ be a fixed basepoint. There is a constant $c(o)>0$
such that, for any increasing diverging function
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ there is some $q\in Q^{1}(o)$ and
diverging increasing sequences $(t_{n})$, $(s_{n})$ such that

$\operatorname{Drift}_{\mu}({\mathcal{R}(q;t_{n})})>c(o)e^{t_{n}}$

and

$\operatorname{Drift}_{\mu}({\mathcal{R}(q;s_{n})})<f(s_{n})s_{n}.$

That is, there are directions along which the growth of the drift is
exponential along one subsequence, while almost linear along another.

We also show that the quasiconvexity of hyperbolic lengths along geodesics
proven by Lenzhen–Rafi [LR11] can be adapted to the drift.

######  Corollary 1.4.

Let $\mu$ be an admissible measure on $\pi_{1}(S,p)$ with finite first moment.
There exists a constant $K^{\prime}>0$ such that for any geodesic $G$ and
points $x,y,z\in\mathcal{T}(S)$ appearing in that order along $G$, we have

$\operatorname{Drift}_{\mu}(y)\leq
K^{\prime}\max(\operatorname{Drift}_{\mu}(x),\operatorname{Drift}_{\mu}(z)).$

The drift of the random walk is closely related with the dimension of the
hitting measure. Let $\mathbb{H}$ be the hyperbolic plane. Each metric $\rho$
can be interpreted as a representation
$\rho:\pi_{1}(S,p)\to\operatorname{Isom}_{+}(\mathbb{H})$, which can then be
extended to an action of $\pi_{1}(S,p)$ on the Gromov boundary
$\partial\mathbb{H}$. We then define the measure $\mu_{\rho}$ on
$\operatorname{Isom}_{+}(\mathbb{H})$ as the pushforward measure of $\mu$ by
$\rho$. A measure $\nu\in\partial\mathbb{H}$ is $\mu_{\rho}$-stationary if for
every measurable $A\subset\partial\mathbb{H}$ we have

$\nu(A)=\sum_{g\in\rho(\pi_{1}(S,p))}\mu(\rho^{-1}(g))\nu(g^{-1}A).$

The measure $\nu$ can also be interpreted as the hitting measure of a random
walk on $\mathbb{H}$ driven by $\mu_{\rho}$. Each $\mu_{\rho}$ has a unique
stationary measure, which we denote by $\nu_{\rho}$. Let
$\dim(\nu_{\rho})\in[0,1]$ denote the Hausdorff dimension of $\nu_{\rho}$.
Deroin–Kleptsyn–Navas [DKN09, Conjecture 1.21], and Karlsson–Ledrappier [KL11]
more generally, conjectured that the dimension is bounded away from 1 whenever
the support of $\mu$ is finite. Kosenko–Tiozzo [KT22] and
Azemar–Gadre–Gouëzel–Haettel–Lessa–Uyani [AGG+nt] recently made some partial
progress regarding this conjecture, but it is still open.

Tanaka [Tan19] shows that

$\dim(\nu_{\rho})=\frac{h}{\operatorname{Drift}_{\mu}(\rho)},$

where $h$ is the entropy of the random walk, a value depending only on the
measure $\mu$. In view of this relation, each result regarding the value of
the drift can be translated to a result regarding the dimension of the
stationary measure. For example, from Theorem 1.2 we have the following.

######  Corollary 1.5.

Let $\mu$ be an admissible measure on $\pi_{1}(S,p)$ with finite first moment
and finite entropy. Furthermore, let $o\in\mathcal{T}(S)$ be a basepoint,
$0<\theta<1$ and $\lambda$ be the Lebesgue measure on $Q^{1}(o)$. Then, for
$\lambda$-almost all directions $q\in Q^{1}(o)$ we have

$e^{-\frac{1}{\theta}t}<\dim(\nu_{{\mathcal{R}(q;t)}})<e^{-\theta t},$

for any $t>t(q)$, where $t(q)$ is some finite time depending on $q$.

###  1.1 Proof overview

Azemar–Gadre–Gouëzel–Haettel–Lessa–Uyani [AGG+nt] proved that the drift of the
random walk is, up to a multiplicative constant, greater than the maximum
length of a fixed finite filling set of curves. The driving observation of
this paper is that this inequality can be improved to an equality up to a
constant multiple. That is, we first prove the following result.

######  Theorem 1.6.

Let $\mu$ be an admissible measure on $\pi_{1}(S,p)$ with finite first moment.
Furthermore, let $F$ be a finite filling set of curves in $S$ and let
$M_{\rho}^{F}$ be the maximum hyperbolic length among the curves with respect
to the hyperbolic metric $\rho$. Then there is some $K<\infty$ depending
solely on $\mu$ and $F$ such that

$\frac{1}{K}M_{\rho}^{F}\leq\operatorname{Drift}_{\mu}(\rho)\leq
KM_{\rho}^{F}.$

Many properties already known about the behaviour of hyperbolic lengths can
then be translated directly to the behaviour of the drift. Of particular
relevance are the works of Masur [Mas82], Choi–Rafi [CR07] and Lenzhen–Rafi
[LR11], who, respectively, find values for the growth of the hyperbolic
lengths along Jenkins–Strebel directions, show relations for the length of
hyperbolic curves between thick parts of space, and prove the quasiconvexity
of the hyperbolic length up to a constant.

Choi–Rafi’s estimates of the hyperbolic lengths in the thick part, combined
with Lenzhen–Rafi’s quasi-convexity can be used to obtain exponential growth
along the whole geodesic, provided the said geodesic does not spend too much
proportional continuous time outside the thick part, relative to the total
time from the start. The control on the continuous time spent outside the
thick part by a common geodesic is given by estimates of Dowdall–Duchin–Masur
[DDM14] in the case of many Lebesgue-like measures, and of
Azemar–Gadre–Jeffreys [AGJ22] for harmonic measures arising from random walks
on the mapping class group with finite first moments.

Finally, we use that both Jenkins–Strebel and recurrent directions are dense
to build a geodesic which alternates between approaching the linear growth of
Jenkins–Strebel directions and the exponential growth of recurrent directions.

### Acknowledgments

The author would like to thank Maxime Fortier Bourque and Vaibhav Gadre for
many helpful discussions and corrections.

##  2 Background

###  2.1 space

In this section we fix the notation and introduce some concepts needed for the
rest of the paper. A detailed introduction to theory can be found, among
others places, in the work of Farb–Margalit [FM12] or Gardiner [GL00]. Let $S$
be a closed surface with negative Euler characteristic. The _space_ of $S$ is
defined as the set of equivalence classes of pairs $(X,f)$ where $X$ is a
Riemann surface and $f:S\to X$ is an orientation preserving homeomorphism. Two
pairs $(X,f)$ and $(Y,g)$ are equivalent if there is a conformal
diffeomorphism $h:X\to Y$ such that $g^{-1}\circ h\circ f$ is isotopic to the
identity.

The _distance_ between two points $[(X,f)],[(Y,g)]\in\mathcal{T}(S)$ is
defined by $d_{\operatorname{Teich}}([(X,f)],[(Y,g)])=\frac{1}{2}\log\inf K$,
where the infimum is taken over all $K\geq 1$ such that there exists a
$K$-quasiconformal homeomorphism $h:X\to Y$ with $g^{-1}\circ h\circ f$
isotopic to the identity. There is a quasiconformal map realizing the infimum.

Let $\gamma$ be an isotopy class of an essential curve in $S$ and
$[(X,f)]\in\mathcal{T}(S)$. We shall usually refer to curves when we actually
mean their isotopy classes. By the uniformization theorem there is a unique
hyperbolic metric on $X$. We define the hyperbolic length of $\gamma$ in
$[(X,f)]$ as the shortest length of the isotopy class $f(\gamma)$ in the
unique hyperbolic metric on $X$. We shall denote this value as
$\text{Hyp}_{[(X,f)]}(\gamma)$.

Given an $\varepsilon>0$ the _$\varepsilon$ -thick_ part of space
$\mathcal{T}_{\varepsilon}(S)\subset\mathcal{T}(S)$ is the subset of marked
hyperbolic metrics where all essential closed curves have hyperbolic length at
least $\varepsilon$.

Let $TX$ be the tangent bundle over $X$. A _quadratic differential_ of a
Riemann surface $X$ is a map $q:TX\to\mathbb{C}$ such that $q(\lambda
v)=\lambda^{2}q(v)$ for every $\lambda\in\mathbb{C}$ and $v\in TX$. We denote
by $Q([(X,f)])$ the holomorphic quadratic differentials with finite area under
the conformal structure of $[(X,f)]$. There is a natural identification
between $Q([(X,f)])$ and the cotangent space at $[(X,f)]$, so for each $q\in
Q([(X,f)])$ we denote as
${\mathcal{R}(q;\cdot)}:\mathbb{R}_{+}\to\mathcal{T}(S)$ the geodesic
associated to $q$. If we restrict to quadratic differentials of area $1$,
which we denote as $Q^{1}([(X,f)])$, the associated geodesics have speed $1$.
That is, for $q\in Q^{1}([(X,f)])$,
$d_{\operatorname{Teich}}({\mathcal{R}(q;t)},{\mathcal{R}(q;s)})=|t-s|$. Given
a quadratic differential $q$ we can define its vertical foliation $V(q)$ as
the measured foliation defined by the smooth paths $\gamma$ such that
$q(\gamma^{\prime}(t))<0$ for every $t$ in the interior of the domain, as well
as the transverse measure given by $|\text{Re}\sqrt{q}|$. If each non critical
trajectory is a closed curve the quadratic differential is called
_Jenkins–Strebel_. The set of Jenkins–Strebel quadratic differentials is dense
within the set of differentials.

By considering the sphere of radius one around the point $o$ we get an
embedding of $Q^{1}(o)$ into $\mathcal{T}(S)$, so we can endow $Q^{1}(o)$ with
a Lebesgue measure. On the other hand, Kaimonovich–Masur [KM96, Theorem 2.2.4]
proved that for non-elementary random walks on the mapping class group,
typical sample paths converge to the Tursthon boudnary of space and that
resulting stationary measure is supported on uniquely ergodic measured
geodesics. As proven by Masur [Mas82] differentials in $Q^{1}(o)$ with
uniquely ergodic vertical foliations converge to the projective class of that
foliation in the Thurston boundary, Hence, the stationary measure can be
pulled back to get another measure on $Q^{1}(o)$.

###  2.2 Random walks

Let $\Gamma$ be a group and $\mu$ a probability measure on $\Gamma$. We say
that $\mu$ is _admissible_ if the semi group generated by its support is equal
to $\Gamma$. We shall always assume that $\mu$ is admissible. The random walk
on $\Gamma$ determined by $\mu$ is the sequence of random variables
$(Z_{n})_{n\in\mathbb{N}}$ defined by

$Z_{n}=g_{1}g_{2}\ldots g_{n},$

where $g_{i}$ are $i.i.d.$ elements of $\Gamma$ with distribution $\mu$.

Let $(\mathbb{H},d)$ be the hyperbolic plane, and
$G=\operatorname{Isom}_{+}(\mathbb{H})$ be the group of orientation preserving
isometries of $\mathbb{H}$. Furthermore, let $\Gamma=\pi_{1}(S,p)$ be the
fundamental group of $S$ based at $p$ and let $\rho:\Gamma\to G$ be a discrete
faithful representation. The length of an element $g\in\Gamma$ with respect to
$\rho$ is defined $|g|_{\rho}=\inf_{x\in\mathbb{H}}d(x,\rho(g)x)$.
Equivalently, $|g|_{\rho}$ is the length of the core curve of the hyperbolic
cylinder $\mathbb{H}/\rho(g)$, or the minimal length within the isotopy class
of curves associated to $\rho(g)$ in $\mathbb{H}/\rho(\Gamma)$. We say $\mu$
has _finite first moment_ with respect to $\rho$ if
$\sum_{g\in\Gamma}|g|_{\rho}\mu(g)<\infty$. We shall always assume that $\mu$
has finite first moment with respect to some $\rho$. The _linear drift_ of the
random walk is then defined as

$\operatorname{Drift}_{\mu}(\rho):=\lim_{n\to\infty}\frac{|Z_{n}|_{\rho}}{n}.$

By Kingman’s subadditive ergodic theorem the drift is well-defined, and since
the measure has finite first moment with respect to $\rho$, it is finite.

For each point $[(X,f)]\in\mathcal{T}(S)$ we can choose a faithful
representation $\rho:\Gamma\to G$ such that $\mathbb{H}/\rho(\Gamma)=X$ and
$\rho(\gamma)$ is the deck transformation associated to $f(\gamma)$. While
such assignment is not unique, the length of an element $\gamma\in\Gamma$ is a
well defined function of this assignment, and hence so is
$\operatorname{Drift}_{\mu}(\rho)$. Indeed, for suitably chosen representation
the curve associated to $\rho(\gamma)$ is $f(\gamma)$, so
$|\gamma|_{\rho}=\text{Hyp}_{[(X,f)]}(\gamma)$. Therefore the elements of
$\mathcal{T}(S)$ can be seen as classes of discrete faithful representations
$\rho:\Gamma\to G$. By a slight abuse of notation we shall treat the elements
of $\mathcal{T}(S)$ as any choice from each class, as the properties we shall
deal with are invariant within the class.

Let $F$ be a fixed finite filling set of closed curves on $S$. Even though we
are asking that the support of $\mu$ generates $\Gamma$ as a semigroup, it
might be that $F$ is not contained entirely in the support of $\mu^{k}$ for
any $k\geq 1$, as the elements of the support of $\mu^{k}$ are formed by
concatenating precisely $k$ elements from the support of $\mu$. To avoid this
we add the neutral element to the support of $\mu$ as follows. Let
$\varepsilon>0$ and let
$\mu_{\varepsilon}=(1-\varepsilon)\mu+\varepsilon\delta_{e}$ be a relaxation
of $\mu$, where we introduce a slight probability of the random walk not
moving at each step. For any $\rho\in\mathcal{T}(S)$ we have
$\operatorname{Drift}_{\mu_{\varepsilon}}(\rho)=(1-\varepsilon)\operatorname{Drift}_{\mu}(\rho)$.
Furthermore, since $\mu$ is admissible, there is some $k\in\mathbb{N}$ such
that the curves associated to the group elements of
$\operatorname{supp}(\mu_{\varepsilon}^{k})$ contain $F$. Then,
$\operatorname{Drift}_{\mu_{\varepsilon}^{k}}(\rho)=k(1-\varepsilon)\operatorname{Drift}_{\mu}(\rho)$.
Since all the results in this paper regarding the drift are true up to
multiplicative constants we will assume that $\operatorname{supp}(\mu)$
already contains $F$. Denote $M_{\rho}^{F}=\max_{\gamma\in
F}(\text{Hyp}_{\rho}(\gamma))$. We shall first prove the following well known
result. The proof is similar to the one done by Minsky [Min93, Lemma 4.7]

######  Lemma 2.1.

Let $F$ be a finite filling set of curves in $S$ and let $o\in\mathcal{T}(S)$
be a fixed basepoint. Then, there is a constant $K$ such that, for any
$\rho\in\mathcal{T}(S)$ and free curve $\gamma$ in $S$ we have,

$\text{Hyp}_{\rho}(\gamma)\leq KM_{\rho}^{F}\text{Hyp}_{o}(\gamma).$

###### Proof.

Let $D_{\rho}$ be the maximum diameter of the polygons formed by the distance
minimizing configuration of the curves $F$, and $\partial D_{\rho}$ its
perimeter. Since the curves of $F$ cut $S$ into polygons, the value of
$\text{Hyp}_{\rho}(\gamma)$ is bounded by the number of intersections of the
curve $\gamma$ with $F$ multiplied by the maximum diameter of the polygons.
That is,

$\text{Hyp}_{\rho}(\gamma)\leq D_{\rho}\sum_{\alpha\in
F}i(\gamma,\alpha)\leq\partial D_{\rho}\sum_{\alpha\in F}i(\gamma,\alpha).$

Under the metric defined by $o$, for any closed curve $\alpha$ we can
isometrically embed an annulus around $\alpha$ of thickness
$\delta_{o}(\alpha)>0$. Then, for any other curve $\gamma$ we have
$i(\alpha,\gamma)\delta_{o}(\alpha)\leq\text{Hyp}_{o}(\gamma)$. Hence,

$\text{Hyp}_{\rho}(\gamma)\leq\partial D_{\rho}\sum_{\alpha\in
F}\frac{1}{\delta(\alpha)}\text{Hyp}_{o}(\gamma).$

Finally, the maximum perimeter of the polygons is smaller than twice the sum
of the lengths of all the curves in $F$, so, denoting by $|F|$ the cardinality
of $F$,

$\partial D_{p}\leq 2\sum_{\alpha\in F}\text{Hyp}_{p}(\alpha)\leq
M^{F}_{p}2|F|.$

We get the result by setting $K=2|F|\sum_{\alpha\in
F}\frac{1}{\delta(\alpha)}$. ∎

Since the number of curves in $F$ is finite, $M_{\rho}^{F}$ is finite for all
$\rho\in\mathcal{T}(S)$. Hence, for any other $\rho\in\mathcal{T}(S)$ we have

$\sum_{g\in\Gamma}|g|_{\rho}\mu(g)\leq
KM_{\rho}^{F}\sum_{g\in\Gamma}|g|_{o}\mu(g)<\infty.$

Therefore, if $\mu$ has finite first moment with respect so some basepoint
$o\in\mathcal{T}(S)$, it has finite first moment for all
$\rho\in\mathcal{T}(S)$. That is, it makes sense to say that the measure $\mu$
has finite first moment if it has finite first moment with respect to at least
one point in $\mathcal{T}(S)$.

We have the following lower bound.

######  Theorem 2.2 (Azemar–Gadre–Gouëzel–Haettel–Lessa–Uyani [AGG+nt, Theorem
3.3]).

Let $\mu$ be an admissible measure on $\pi_{1}(S,p)$ with finite first moment,
and let $F$ be a finite filling set of curves in $S$. There exists a constant
$c>0$ such that $\operatorname{Drift}_{\mu}(\rho)\geq cM_{\rho}^{F}$ for all
$[\rho]\in\mathcal{T}(S)$.

Using Lemma 2.1 it is relatively straightforward to check that the value
$M_{\rho}^{F}$ also serves to give an upper bound.

###### Proof of Theorem 1.6.

By Theorem 2.2 there is $c>0$ such that $\operatorname{Drift}_{\mu}(\rho)\geq
cM_{\rho}^{F}$ , so we only have to prove the upper bound. Fix some basepoint
$o\in\mathcal{T}(S)$. By Lemma 2.1 there is some $C<\infty$ such that
$|Z_{n}|_{\rho}\leq CM_{p}|Z_{n}|_{o}$. Therefore,

$\operatorname{Drift}_{\mu}(\rho)=\lim_{n\to\infty}\frac{|Z_{n}|_{\rho}}{n}\leq
CM_{p}\lim_{n\to\infty}\frac{|Z_{n}|_{o}}{n}=CM_{p}\operatorname{Drift}_{\mu}(o),$

so the result follows from setting
$K=\max\left(C\operatorname{Drift}_{\mu}(o),\frac{1}{c}\right)$ ∎

##  3 Linear growth

We recall that the growth has to be at least linear.

######  Theorem 3.1 (Azemar–Gadre–Gouëzel–Haettel–Lessa–Uyani).

Given a basepoint $o\in\mathcal{T}(S)$ there exists some constant $c$ such
that $\operatorname{Drift}_{\mu}(\rho)\geq cd_{\operatorname{Teich}}(\rho,o)$.

In this section we prove that this bound is sharp. The main ingredient is the
following result, established by Masur [Mas82, End of the proof of Theorem
1.1], which finds limiting values for hyperbolic lengths along Jenkins–Strebel
quadratic differentials.

######  Theorem 3.2 (Masur).

Let $q$ be a unit area Jenkins–Strebel quadratic differential and let
$\alpha_{1},\ldots,\alpha_{k}$ be its core curves. Then, for any sequence
$(\rho_{n})\subset\mathcal{T}(S)$ converging to $q$ in the visual
compactification and any curve $\gamma$ in $S$ we have

$\lim_{n\to\infty}\frac{\text{Hyp}_{\rho_{n}}(\gamma)}{4d_{\operatorname{Teich}}(o,\rho_{n})}=\sum_{i=1}^{k}i(\alpha_{i},\gamma).$

###### Proof of Theorem 1.1.

By Theorem 3.1 we only have to prove the upper bound. Let
$\alpha_{1},\ldots,\alpha_{k}$ be the core curves of the vertical foliation of
$q$ and let $\gamma_{1},\ldots,\gamma_{l}$ be the curves within the filling
set $F$. Let $\delta>0$. Then, by Theorem 3.2 for each $\gamma_{j}$ we have a
$t_{j}$ such that, for all $t>t_{j}$, we have

$\text{Hyp}_{{\mathcal{R}(q;t)}}(\gamma_{j})<4\left(\sum_{i=1}^{k}i(\alpha_{i},\gamma_{j})+\delta\right)t.$

Hence we can take $C(q,j)$ big enough so
$\text{Hyp}_{{\mathcal{R}(q;t)}}(\gamma_{j})<C(q,j)t$ for all $t\geq 1$. The
theorem follows by setting $C(q)=K\max_{j\leq k}C(q,j)$, where $K$ is the
constant given by Theorem 1.6. ∎

##  4 Exponential growth

The goal of this section is to prove that the standard behaviour of the drift
is exponential growth with respect to the distance. We begin by observing that
as a direct result of Wolpert’s Lemma, the growth can not be higher than
exponential.

######  Proposition 4.1.

Given a basepoint $o\in\mathcal{T}(S)$, there exists some constant $C(o)$ such
that
$\operatorname{Drift}_{\mu}(\rho)\leq\operatorname{Drift}_{\mu}(o)e^{2d_{\operatorname{Teich}}(\rho,o)}$.

###### Proof.

As proven by Wolpert [Wol79, Lema 3.1] for any two points
$o,\rho\in\mathcal{T}(S)$ and loop $\gamma$ we have

$\text{Hyp}_{\rho}(\gamma)\leq
e^{2d_{\operatorname{Teich}}(o,\rho)}\text{Hyp}_{o}(\gamma).$

Therefore,

$\operatorname{Drift}_{\mu}(\rho)=\lim_{n\to\infty}\frac{\text{Hyp}_{\rho}(Z_{n})}{n}\leq
e^{2d_{\operatorname{Teich}}(o,\rho)}\lim_{n\to\infty}\frac{\text{Hyp}_{o}(Z_{n})}{n}=e^{2d_{\operatorname{Teich}}(o,\rho)}\operatorname{Drift}_{\mu}(o).$

∎

The following result by Choi–Rafi [CR07, Theorem B] allows us to improve the
previous upper bound, as well as get a lower bound for the growth of the drift
for points in the thick part of space. Recall that
$\mathcal{T}_{\varepsilon}(S)$ denotes the $\varepsilon$-thick part of space.

######  Theorem 4.2 (Choi–Rafi).

Fix $\varepsilon>0$ and $o\in\mathcal{T}_{\varepsilon}(S)$. There is a finite
filling set of closed curves $G$ and a constant $D>0$ such that, for any
$\rho\in T_{\varepsilon}(S)$ we have

$\left|d_{\operatorname{Teich}}(\rho,o)-\log\left(\max_{\alpha\in
G}\frac{\text{Hyp}_{\rho}(\alpha)}{\text{Hyp}_{o}(\alpha)}\right)\right|\leq
D.$

######  Proposition 4.3.

Let $\varepsilon>0$ and let $o\in\mathcal{T}_{\varepsilon}(S)$. Then, there
exists constants $c,C>0$ such that for any
$\rho\in\mathcal{T}_{\varepsilon}(S)$ we have

$ce^{d_{\operatorname{Teich}}(\rho,o)}\leq\operatorname{Drift}_{\mu}(\rho)\leq
Ce^{d_{\operatorname{Teich}}(\rho,o)}.$

###### Proof.

Let $G$ be the filling set of curves from Theorem 4.2. For any curve
$\gamma\in G$ we have

$\text{Hyp}_{\rho}(\gamma)=\text{Hyp}_{o}(\gamma)\frac{\text{Hyp}_{\rho}(\gamma)}{\text{Hyp}_{o}(\gamma)}\leq\text{Hyp}_{o}(\gamma)\max_{\alpha\in
G}\frac{\text{Hyp}_{\rho}(\alpha)}{\text{Hyp}_{o}(\alpha)}.$

Then, taking the maximum among all $\gamma\in G$ in the previous inequality we
have, by Theorem 1.6, some $K>0$ such that

$\operatorname{Drift}_{\mu}(\rho)\leq K\max_{\alpha\in
G}{\text{Hyp}_{\rho}(\alpha)}\leq K\max_{\alpha\in
G}\text{Hyp}_{o}(\alpha)e^{D}e^{d_{\operatorname{Teich}}(\rho,o)},$

where the last inequality follows from applying Theorem 4.2. On the other
hand,

$\frac{\max_{\alpha\in G}{\text{Hyp}_{\rho}(\alpha)}}{\min_{\alpha\in
G}\text{Hyp}_{o}(\alpha)}\geq\max_{\alpha\in
G}\frac{\text{Hyp}_{\rho}(\alpha)}{\text{Hyp}_{o}(\alpha)},$

so similarly we have

$\operatorname{Drift}_{\mu}(\rho)\geq\frac{1}{K}\max_{\alpha\in
G}{\text{Hyp}_{\rho}(\alpha)}\geq\frac{1}{K}\min_{\alpha\in
G}\text{Hyp}_{o}(\alpha)e^{-D}e^{d_{\operatorname{Teich}}(\rho,o)}.$

Hence, the result follows from setting $c=\frac{1}{K}\min_{\alpha\in
G}(\text{Hyp}_{o}(\alpha))e^{-D}$ and $C=K\max_{\alpha\in
G}(\text{Hyp}_{o}(\alpha))e^{D}$. ∎

By Mumford’s compactness the preimage of every bounded subset of moduli space
is contained in some $\varepsilon$-thick part of space for $\varepsilon$ small
enough. Hence we have the following result.

######  Corollary 4.4.

Let $\gamma:[0,\infty)\to\mathcal{T}(S)$ be a ray such that its image is
bounded in moduli space. Then, there is $c,C>0$ such that

$ce^{t}\leq\operatorname{Drift}_{\mu}(\gamma(t))\leq Ce^{t}.$

To prove that for almost all directions the growth is exponential we first
observe that the growth of the drift is, up to a constant, not lost for big
enough times. The main ingredient we shall use in the proof is the following.

######  Theorem 4.5 (Lenzhen–Rafi [LR11, Theorem A]).

There exists a constant $K>0$ such that for every closed curve $\gamma$, any
geodesic $G$ and points $x,y,z\in\mathcal{T}(S)$ appearing in that order along
$G$, we have

$\text{Hyp}_{y}(\gamma)\leq
K\max(\text{Hyp}_{x}(\gamma),\text{Hyp}_{z}(\gamma)).$

By Theorem 1.6 Lenzhen–Rafi’s result translates directly to the drift. That
is, we have Corollary 1.4.

###### Proof of Corollary 1.4.

By Theorems 1.6 and 4.5 there are constants $C,K>0$ such that

$\operatorname{Drift}_{\mu}(y)\leq C\max_{\gamma\in
F}\text{Hyp}_{y}(\gamma)\leq KC\max_{\gamma\in
F}\left(\max(\text{Hyp}_{x}(\gamma),\text{Hyp}_{z}(\gamma)\right)$

Switching the order of the maximums we have

$\operatorname{Drift}_{\mu}(y)\leq KC\max\left(\max_{\gamma\in
F}\text{Hyp}_{x}(\gamma),\max_{\gamma\in F}\text{Hyp}_{z}(\gamma)\right)\leq
KC^{2}\max(\operatorname{Drift}_{\mu}(x),\operatorname{Drift}_{\mu}(z)).$

∎

Furthermore, the drift is a proper function, so along any ray
${\mathcal{R}(q;\cdot)}$, the value of
$\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})$ is eventually larger than
$\operatorname{Drift}_{\mu}({\mathcal{R}(q;0)}$. Hence, we have the following.

######  Lemma 4.6.

There is some constant $K>0$ such that, for any basepoint $o\in\mathcal{T}(S)$
there is some time $t_{o}$ for which

$\operatorname{Drift}_{\mu}({\mathcal{R}(q;s)})\leq
K\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})$

for any $t>t_{o}$, $t>s>0$ and $q\in Q^{1}(o)$.

###### Proof.

Let $t_{o}$ be big enough so
$\operatorname{Drift}_{\mu}(\rho)>\operatorname{Drift}_{\mu}(o)$ for any
$\rho$ such that $d_{\operatorname{Teich}}(\rho,o)>t_{o}$. By Theorem 3.1 such
a $t_{o}$ exists. Then, for any quadratic differential $q$ based at $o$,
$t>t_{o}$ and $t>s>0$ we have, by Corollary 1.4,

$\operatorname{Drift}_{\mu}({\mathcal{R}(q;s)})\leq
K\max\left(\operatorname{Drift}_{\mu}({\mathcal{R}(q;0)},\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)}))\right)=K\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})$

∎

Lemma 4.6, combined with Proposition 4.3, gives upper and lower bound on the
growth of the drift along a geodesic provided said geodesic does not spend too
much continued time outside the thick part. For a given geodesic
${\mathcal{R}(q;\cdot)}$, $\varepsilon>0$ and $t>0$, we aim to find some
control on the largest $s^{bot}_{q}(t)\leq t$ and lowest $s^{top}_{q}(t)\geq
t$ such that
${\mathcal{R}(q;s^{bot}_{q}(t))},{\mathcal{R}(q;s^{top}_{q}(t))}\in\mathcal{T}_{\varepsilon}(S)$.
Given a geodesic ray $\gamma:\mathbb{R}_{+}\to\mathcal{T}(S)$, define the
proportion of the amount of time spent in the thick part up to time $t$ as

$\operatorname{Thick}_{\varepsilon}^{\%}(\gamma,t):=\frac{|\\{0\leq s\leq
t\,:\,\gamma(s)\in\mathcal{T}_{\varepsilon}(S)\\}|}{t}.$

######  Theorem 4.7.

Let $o\in\mathcal{T}(S)$ be a basepoint in space. Furthermore, let $\sigma$ be
a measure on $Q^{1}(o)$ such that for all $0<\xi<1$ there is some
$\varepsilon(\xi)>0$ such that for $\sigma$-almost every $q$ there is
$t_{q}^{\xi}<\infty$ such that

$\operatorname{Thick}_{\varepsilon(\xi)}^{\%}({\mathcal{R}(q;\cdot)},t)\geq\xi$

for all $t\geq t^{\xi}_{q}$.

Fix then $0<\theta<1$. For $\sigma$-almost all directions $q\in Q^{1}(o)$
there is $t(q,\theta)<\infty$ such that

$e^{\theta
t}<\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})<e^{\frac{1}{\theta}t}$

for all $t>t(q,\theta)$.

###### Proof.

Given $0<\theta<1$, let $\xi=\frac{1+\theta}{2}$. For a given $q\in Q^{1}(o)$
and $t>0$ let $s^{top}_{q}(t)\geq t$ be the smallest time larger than $t$ such
that ${\mathcal{R}(q;s^{top}_{q}(t))}\in\mathcal{T}_{\varepsilon(\xi)}(S)$.
The time spent outside $\mathcal{T}_{\varepsilon(\xi)}(S)$ directly after $t$
is $s^{top}_{q}(t)-t$. Hence,
$\frac{s^{top}_{q}(t)-(s^{top}_{q}(t)-t)}{s^{top}_{q}(t)}>\xi$. Therefore,
$s^{top}_{q}(t)<\frac{1}{\xi}t$. Hence, By Lemma 4.6 we have $K,t_{o}>0$ such
that, for all $t>t_{o}$,

$\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})\leq
K\operatorname{Drift}_{\mu}({\mathcal{R}(q;s^{top}_{q}(t))}).$

Since ${\mathcal{R}(q;s^{top}_{q}(t))}$ is in the $\varepsilon$-thick part of
space we have, by Proposition 4.3,

$\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})\leq
K\operatorname{Drift}_{\mu}({\mathcal{R}(q;s^{top}_{q}(t))}\leq
CKe^{s^{top}_{q}(t)}\leq CKe^{\frac{1}{\xi}t}.$

Similarly, denoting $s^{bot}_{q}(t)$ the largest time smaller than $t$ such
that ${\mathcal{R}(q;s^{top}_{q}(t))}\in\mathcal{T}_{\varepsilon(\xi)}(S)$ we
get $s^{bot}_{q}(t)>\xi t$. Following the same reasoning we get

$\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})\geq\frac{1}{K}\operatorname{Drift}_{\mu}({\mathcal{R}(q;s^{bot}_{q}(t))}\geq\frac{c}{K}e^{s^{bot}_{q}(t)}\geq\frac{c}{K}e^{\xi
t}.$

Since $\xi<\theta<1$ there is some $t_{\theta}$ such that $e^{\theta
t}>\frac{c}{K}e^{\xi t}$ and $e^{\frac{1}{\theta}t}<CKe^{\frac{1}{\xi}t}$ for
all $t\geq t_{\theta}$. Hence, the theorem follows from setting
$t(q,\theta)=\max(t_{q},t_{\theta},t_{o})$. ∎

It follows from Downdall–Duchin–Masur [DDM14, Proposition 5.5] that the
hypothesis of Theorem 4.7 is satisfied a wide variety of Lebesgue-class
measures, giving us a proof of Theorem 1.2. Furthermore, from
Azemar–Gadre–Jeffreys [AGJ22, Proposition 3.3] the property is also satisfied
for harmonic measures generated by non-elementary measures on the mapping
class group with finite first moment.

######  Corollary 4.8.

Let $\sigma$ be a measure on $Q^{1}(o)$ satisfying the hypothesis of Theorem
4.7. Then, for any $\varepsilon>0$ we have, for $\sigma$-almost every $q\in
Q^{1}(o)$,

$\liminf_{t\to\infty}\frac{\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})}{e^{(1-\varepsilon)t}}=\infty$

and

$\limsup_{t\to\infty}\frac{\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})}{e^{(1+\varepsilon)t}}=0.$

Furthermore, there is $\sigma$-almost surely some $0<c(q)<\infty$ such that

$\liminf_{t\to\infty}\frac{\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})}{e^{t}}\leq
c(q)\leq\limsup_{t\to\infty}\frac{\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})}{e^{t}}.$

###### Proof.

Let $\theta=1-\varepsilon$. Then, by Theorem 1.2 there is some $c>0$ such that
$\sigma$-almost every $q\in Q^{1}(o)$ we have

$\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})\geq e^{(1-\varepsilon)t}$

for all $t$ big enough. Hence, the first relation follows. Similarly for
$\theta=\frac{1}{1+\varepsilon}$ we get the second relation. The last relation
follows from taking a diverging sequence of times $(t_{n})$ such that
${\mathcal{R}(q;t_{n})}\in\mathcal{T}_{\varepsilon}(S)$ and applying
Proposition 4.3. ∎

##  5 Variable growth

In this section we prove that there is some geodesic along which we have
variable growth. The basic idea of the proof is alternating Proposition 4.3
and Theorem 1.1, using the fact that both results apply to dense sets of
directions.

###### Proof of Theorem 1.3.

Fix $\delta,\varepsilon>0$ such that the basepoint $o\in\mathcal{T}(S)$ is in
the $\varepsilon$-thick part of space. We shall build inductively a sequence
of quadratic differentials $(j_{k})$, as well as sequences of times $(s_{k})$,
$(t_{k})$ such that,
$\operatorname{Drift}_{\mu}({\mathcal{R}(j_{k};s_{i})})<\left(f(s_{i})-\delta
2^{-2(k-i)}\right)s_{i}$ for each $i\leq k$ and each
${\mathcal{R}(j_{k};t_{i})}$ is at most at distance
$\delta\left(2-2^{-2(k-i)}\right)$ from $\mathcal{T}_{\varepsilon}$ for each
$i\leq k$. The theorem will follow by taking an accumulation point of such
sequence.

For $k=0$ let $j_{0}\in Q^{1}(o)$ be a Jenkins-Strebel quadratic differential.
By Theorem 1.1 there are constants $C(j_{0}),t(j_{0})$ such that
$\operatorname{Drift}_{\mu}({\mathcal{R}(j_{0};t)})<C(j_{0})t$ for all
$t>t(j_{0})$. Let $s_{0}$ be big enough such that $f(s_{0})>C(j_{0})+\delta$
and $s_{0}>t(j_{0})$. Finally, let $t_{0}=0$.

Assume then we have the sequence up to $k$. The set of recurrent directions to
$\mathcal{T}_{\varepsilon}$ is dense, so we can take a sequence of quadratic
differentials $(q^{n})\subset Q^{1}(o)$ spawning recurrent geodesics and
converging to $j_{k}$. Since $q^{n}\to j_{k}$, the geodesics
${\mathcal{R}(q^{n};\cdot)}$ converge to the geodesic
${\mathcal{R}(j_{k};\cdot)}$ pointwise. The drift is a continuous function
with respect to space, so
$\operatorname{Drift}_{\mu}({\mathcal{R}(q^{n};s_{i})})\to\operatorname{Drift}_{\mu}({\mathcal{R}(j_{k};s_{i})})\leq(f(s_{i})-\delta
2^{-2(k-i)})s_{i}$ for each $i\leq k$. Let $q$ be the first element of the
sequence $(q^{n})$ such that

$\operatorname{Drift}_{\mu}({\mathcal{R}(q;s_{i})})<(f(s_{i})-\delta
2^{-(2(k-i)+1)})s_{i}$

and

$d_{\operatorname{Teich}}({\mathcal{R}(q;t_{i})},{\mathcal{R}(j_{k};t_{i})})<\delta
2^{-(2(k-i)+1)}$

for all $i\leq k$. The geodesic ${\mathcal{R}(q;\cdot)}$ is recurrent, so we
can fix $t_{k+1}>s_{k}+1$ such that
${\mathcal{R}(q;t_{k+1})}\in\mathcal{T}_{\varepsilon}$.

The set of Jenkins–Strebel directions is dense, so we can take a sequence
$(j^{n})$ converging to $q$. As before, the convergence within the sequences
is pointwise, so
$\operatorname{Drift}_{\mu}({\mathcal{R}(j^{n};s_{i})})\to\operatorname{Drift}_{\mu}({\mathcal{R}(q;s_{i})})\leq(f(s_{i})-\delta
2^{-(2(k-i)+1)})s_{i}$ for each $i\leq k$. Let $j_{k+1}$ be the first element
of the sequence $(j^{n})$ such that

$\operatorname{Drift}_{\mu}({\mathcal{R}(j_{k+1};s_{i})})\leq(f(s_{i})-\delta
2^{-2(k+1-i)})s_{i}$

and

$d_{\operatorname{Teich}}({\mathcal{R}(j_{k+1};t_{i})},{\mathcal{R}(q;t_{i})})<\delta
2^{-2(k+1-i)}$

for all $i\leq k+1$. As before, there is some $C(j_{k+1})$ such that
$\operatorname{Drift}_{\mu}({\mathcal{R}(j_{k+1};t)})<C(j_{k+1})t$, so let
$s_{k+1}$ be the first time larger than $t_{k+1}$ such that
$f(s_{k+1})>C(s_{k+1})+\delta$. Furthermore, for $i\leq k$

$d_{\operatorname{Teich}}({\mathcal{R}(j_{k+1};t_{i})},\mathcal{T}_{\varepsilon})\leq
d_{\operatorname{Teich}}({\mathcal{R}(j_{k+1};t_{i})},{\mathcal{R}(q;t_{i})})+d_{\operatorname{Teich}}({\mathcal{R}(q;t_{i})},{\mathcal{R}(j_{k};t_{i})})+d_{\operatorname{Teich}}({\mathcal{R}(j_{k};t_{i})},\mathcal{T}_{\varepsilon})$
$<\delta\left(2^{-(2(k-i)+1)}+2^{-2(k+1-i)}+2-2^{-(2(k-i))}\right)=\delta\left(2-2^{-2(k+1-i)}\right)$

and for $i=k+1$ we have, since
${\mathcal{R}(q_{k+1};t_{k+1})}\in\mathcal{T}_{\varepsilon}$,

$d_{\operatorname{Teich}}({\mathcal{R}(j_{k+1};t_{k+1})},\mathcal{T}_{\varepsilon})\leq
d_{\operatorname{Teich}}({\mathcal{R}(j_{k+1};t_{k+1})},{\mathcal{R}(q_{k+1};t_{k+1})})<\delta.$

Hence, we have completed the induction step.

Let $q_{f}$ be an accumulation point of the sequence $(j_{k})$. There is then
a subsequence, relabeled $(j_{k})$ converging to $q_{f}$. By pointwise
convergence,
$\operatorname{Drift}_{\mu}({\mathcal{R}(j_{k};s_{i})})\to\operatorname{Drift}_{\mu}({\mathcal{R}(q_{f};s_{i})}$,
so $\operatorname{Drift}_{\mu}({\mathcal{R}(q_{f};s_{i})}<f(s_{i})s_{i}$ for
each $i$. Furthermore,
${\mathcal{R}(j_{k};t_{i})}\to{\mathcal{R}(q_{f};t_{i})}$, so for each $i$ the
points ${\mathcal{R}(q_{f};t_{i})}$ are at most at distance $2\delta$ from the
$\mathcal{T}_{\varepsilon}$. Therefore, there is some $\varepsilon^{\prime}$
such that ${\mathcal{R}(q_{f};t_{i})}\in T_{\varepsilon^{\prime}}$ for all
$i$. Hence, by Proposition 4.3 there is some $c>0$ such that
$\operatorname{Drift}_{\mu}({\mathcal{R}(q_{f};t_{i})})>ce^{t_{i}}$.
Furthermore, $t_{k+1}>s_{k}+1>t_{k}+1$, so the sequence $(t_{k})$ diverges to
infinity, and so does $(s_{k})$. Finally, the geodesic
${\mathcal{R}(q_{f};\cdot)}$ is recurrent, so by Masur’s criterion [Mas92,
Theorem 1.1] the vertical foliation of $q_{f}$ is uniquely ergodic. ∎

##  6 Conclusions

Theorem 4.7 shows that along a typical geodesic the drift grows exponentially.
However, it does not determine precisely the fluctuations within such
exponential growth. Furthermore, Theorem 1.3 shows that the behaviour of the
drift along a geodesic can vary wildly, so it is natural to ask whether there
is some variation within a typical geodesic.

######  Question 6.1.

Let $\nu$ be a measure on $Q^{1}(o)$ satisfying the hypothesis of Theorem 4.7.
Do we have

$0<\liminf_{t\to\infty}\frac{\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})}{e^{t}}=\limsup_{t\to\infty}\frac{\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})}{e^{t}}<\infty$

$\nu$-almost surely?

Note that by Corollary 4.8 we have
$\limsup_{t\to\infty}\frac{\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})}{e^{t}}>0$
and
$\liminf_{t\to\infty}\frac{\operatorname{Drift}_{\mu}({\mathcal{R}(q;t)})}{e^{t}}<\infty$.
In the proof of Theorem 4.7 we have bounded the quotient between the drift and
the exponential $e^{t}$ by a function depending on the maximal continuous time
spent in the thin part of space up to time $t$. The growth of these maximal
departures may vary differently depending on the measure. On the one hand,
following work of Gadre [Gad17, Lemma 5.5] it is reasonable to conjecture that
the maximal departure grows slightly faster than $\log(t)$ for the Lebesgue
measure. On the other hand, because of exponential decay of subsurface
projections for harmonic measures, it may be expected (though this is still
unproved) that the largest continuous time spent in the thin part for a
harmonic typical geodesic is of the order of $\log(\log(t))$. Therefore, the
answer to the previous question might be different for the Lebesgue and
harmonic measures.

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