Bellis strong stable sets on infinite hyperbolic surfaces

TL;DR

This paper corrects a proof regarding strong stable sets in the unit tangent bundle of hyperbolic surfaces, showing they differ from horocyclic orbits under certain conditions.

Contribution

It provides a corrected proof of Bellis' theorem about the structure of strong stable sets on hyperbolic surfaces, clarifying their divergence from horocyclic orbits.

Findings

Strong stable sets do not coincide with horocyclic orbits for certain vectors.

Construction of geodesic rays winding around infinitely many closed geodesics.

The proof clarifies the behavior of stable sets in hyperbolic geometry.

Abstract

We provide a corrected proof of a theorem of A. Bellis on strong stable sets in the unit tangent bundle of certain hyperbolic surfaces. The theorem states that, for vectors whose geodesic rays encounter arbitrarily short closed geodesics, the strong stable set in the dynamical sense does not coincide with the associated horocyclic orbit. The proof is based on Bellis' idea of constructing geodesic rays that wind around infinitely many closed geodesics.