Exceptional sets for geodesic flows of noncompact manifolds

TL;DR

This paper investigates the size of sets of geodesic flow points in negatively curved manifolds that avoid a specific subset, showing they can have full topological entropy if the subset's entropy is sufficiently small.

Contribution

It establishes conditions under which the limit exceptional set for geodesic flows has full topological entropy, extending understanding of dynamical complexity in noncompact manifolds.

Findings

Limit $A$-exceptional sets have full topological entropy when $A$'s entropy is less than the flow's entropy.

Results apply to invariant compact subsets and submanifolds.

Provides new insights into the structure of geodesic flows on noncompact manifolds.

Abstract

For a geodesic flow on a negatively curved Riemannian manifold $M$ and some subset $A\subset T^1M$, we study the limit $A$-exceptional set, that is the set of points whose $\omega$-limit do not intersect $A$. We show that if the topological $\ast$-entropy of $A$ is smaller than the topological entropy of the geodesic flow, then the limit $A$-exceptional set has full topological entropy. Some consequences are stated for limit exceptional sets of invariant compact subsets and proper submanifolds.