Counterexamples in nonpositive curvature

TL;DR

This paper constructs specific examples of rank one compact and complete surfaces with unique geodesic flow properties, challenging existing assumptions about geodesic behavior in nonpositive curvature.

Contribution

It provides novel examples of rank one surfaces demonstrating atypical geodesic flow dynamics, including non-shadowable recurrent geodesics and non-dense ergodic measures.

Findings

Existence of recurrent geodesics not shadowed by periodic ones

Ergodic measures are not dense in the invariant measure set

Non-transitive geodesic flow despite dense periodic orbits

Abstract

We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. Finally, we give examples of complete rank one surfaces for which the non wandering set of the geodesic flow is connected, the periodic orbits are dense in that set, yet the geodesic flow is not transitive in restriction to its non wandering set.