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First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by Gadre, Maher and Tiozzo Word length statistics for Teichmüller geodesics and singularity of harmonic measure (see also Gadre,Partial sums of excursions along random geodesics and volume asypmtotics...), the authors consider a quantity $E(\gamma,T)$ which they call the total excursion up to time $T$ of the geodesic $\gamma$. To my possibly wrong understanding, it is roughly the total distance a geodesic traveled into the cusp up to time $T$. These two papers are concerned with studying the asymptotics of this quantity for typical (chosen uniformly w.r.t. some natural measures). My questions are as follows:

(1) What kind of important geometric/ number theoretic are known to be affected by this quantity? What is the importance of knowing its asymptotics.

(2) One can define similarly a quantity $E(\gamma,-T)$ to be the same thing but for the geodesic running in inverse time (similar to the double boundary). I found out that there exists a measurable function (can be chosen to be Cadlag) $\kappa(\gamma,T): T_1(M)\times (0,\infty)\to \{\pm1\}$ so that for a typical geodesic w.r.t. the Liouville measure $$\frac{E(\gamma,\kappa(\gamma,T)\cdot T)}{T\log T}\to 1\ \text{as}\ T\to\infty,$$
and $$\frac{E(\gamma,-\kappa(\gamma,T)\cdot T)}{T\log T}\to \infty\ \text{as}\ T\to\infty.$$
The speed of convergence of the second term to infinity is governed by some extreme value statistics. Does a statement of this type have any meaning in hyperbolic geometry. There is an equivalent theorem for the natural extension of the Farey map--- sums of continued fraction digits.

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