Entropy and self-intersection number of geodesic currents on compact hyperbolic surfaces

TL;DR

This paper establishes a quantitative relationship between the entropy of ergodic geodesic currents on compact hyperbolic surfaces, their self-intersection number, and the systole, showing small self-intersection implies small entropy.

Contribution

It provides a new upper bound on the measure-theoretic entropy of ergodic geodesic currents based on their self-intersection number and the surface's systole.

Findings

Small self-intersection number implies small entropy.

Quantitative upper bound on entropy in terms of self-intersection and systole.

Entropy is controlled by geometric and topological features of the surface.

Abstract

Let $X$ be a compact hyperbolic surface of genus $g$, and $C$ a geodesic current on $X$. Denote by $h_X(C)$ the measure-theoretic entropy of $C$ with respect to the geodesic flow. Assume that $C$ is ergodic. In this paper, we establish a quantitative upper bound on $h_X(C)$ in terms of its self-intersection number $i(C,C)$ and the systole of $X$. In particular, we show that small self-intersection number forces small entropy.