# Expansiveness for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature

## TL;DR

This paper investigates the expansive properties of geodesic and horocycle flows on compact negatively curved surfaces, providing new proofs and clarifying their dynamical characteristics in various mathematical senses.

## Contribution

It offers a new proof of geodesic flow expansiveness and characterizes horocycle flow expansiveness in multiple frameworks, clarifying their dynamical behaviors.

## Key findings

- Geodesic flow is expansive in the Bowen-Walters sense.
- Horocycle flow is positive and negative kinematic expensive in Artigue's sense.
- Horocycle flow is expansive in Katok-Hasselblatt sense but not Bowen-Walters expensive.

## Abstract

We study expansive properties for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature. It is well-known that the geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow is positive and negative separating in the sense of Gura. In this paper, we give a new proof of the expansiveness in the sense of Bowen-Walters for the geodesic flow and show that the horocycle flow is positive and negative kinematic expensive in the sense of Artigue as well as expansive in the sense of Katok-Hasselblatt but not expensive in the sense of Bowen-Walters. We also point out that the geodesic flow is neither positive nor negative separating.