On the non-expansiveness of the geodesic flow on surfaces with cusps

TL;DR

This paper demonstrates that on hyperbolic surfaces with cusps, the geodesic flow exhibits complex orbit behavior with uncountably many orbits in neighborhoods, revealing new dynamical phenomena.

Contribution

It introduces the existence of strong stable sets that differ from stable horocycles, highlighting novel dynamical features of geodesic flows on such surfaces.

Findings

Uncountably many geodesic orbits can be found in any neighborhood.

Strong stable sets differ from traditional stable horocycles.

The phenomenon is typical for finite-volume hyperbolic surfaces.

Abstract

We exhibit orbits of the geodesic flow on a hyperbolic surface with at least one cusp such that every tubular neighborhood contains uncountably many distinct geodesic flow orbits. The proof relies on new phenomena, namely the existence of strong stable sets in the dynamical sense that do not coincide with the stable horocycles. When the surface has finite volume, this phenomenon is typical.