Geodesic Currents on Hyperbolic Surfaces: Entropy, Intersection Number, and Equidistribution

Abstract

Let $X$ be a hyperbolic surface of finite volume. In this thesis, we investigate the behavior of closed geodesics on $X$ and their geometric intersections. Specifically, we address the following three questions:

How does the interaction between the intersection number and length of closed geodesics change when the surface $X$ varies? Let us denote the length and geometric intersection number of a pair of closed geodesics on $X$ by $\ell_X(\alpha)$ and $i(\alpha, \beta)$, respectively, and let $$ I(X) := \sup \limits_{\alpha,\beta} \frac{i(\alpha, \beta)}{\ell_X(\alpha)\ell_X(\beta)}, $$ where the supremum is taken over all pairs of closed geodesics. We refer to $I(X)$ as the \emph{interaction strength} of $X$, as it controls the best upper bound on $i(\alpha,\beta)$ in terms of $\ell_X(\alpha)\ell_X(\beta)$. Let $\mathcal{M}_g$ be the moduli space of compact hyperbolic surfaces with genus $g$. Our main result here describes the exact asymptotic behavior of $I(X)$ on $\mathcal{M}g$. To state it, let $\sys(X) := \inf \limits{\alpha} \ell_X(\alpha)$ denotes the systole of $X$. We show that $$ I(X) \sim \frac{1}{2 \sys(X) \log(1/\sys(X))} $$ as $X \to \infty$ in $\mathcal{M}_g$.

How are the intersection points between all pairs of closed geodesics distributed on $X$? It is not difficult to see that these points are dense in $X$. We show that the intersection points of closed geodesics with length $\leq T$ become equidistributed on $X$, as $T\to \infty$. More precisely, consider all the intersection points (including self-intersections) between closed geodesics with length $\leq T$. Let $\mu_T$ be the counting measure on these points (the points are considered with multiplicity). Our main result here shows that the probability measure $\mu_T/|\mu_T|$ converges to the normalized area measure on $X$, as $T\to \infty$. In the non-compact case, we carry out a careful analysis to show there is no escape of mass in the limit.

Can one control the measure-theoretic entropy $h_{\mu}$ of a geodesic current $\mu$ in terms of its self-intersection number $i(\mu,\mu)$? Geodesic currents generalize closed geodesics, and it is known that geodesic currents with zero self-intersection number have zero measure-theoretic entropy. This observation motivates our question. We show that the answer is affirmative when the geodesic current is ergodic. More precisely, we show that when $\mu$ is an ergodic geodesic current with $\ell_X(\mu)=1$, we have $$ h_{\mu}= \mathcal{O} \left( \sqrt{i(\mu,\mu)}\log \biggl(\frac{1}{\sqrt{i(\mu,\mu)}}\biggr) \right). $$ The proof involves topological and combinatorial estimates for the number of closed geodesics with a small number of self-intersections relative to length.