Ergodicity of a surgered flow on unit tangent bundle of hyperbolic surface

TL;DR

This paper constructs a new ergodic flow on the unit tangent bundle of a genus two hyperbolic surface through a Dehn-type surgery, demonstrating ergodicity in the modified region and linking it to linked twist maps.

Contribution

It introduces a novel surgical method to produce ergodic flows on hyperbolic surface bundles, connecting geometric surgery with dynamical ergodicity.

Findings

The surgered flow is ergodic on the surgery region.

The flow projects to a linked twist map with strong shears.

Ergodicity is achieved through specific geometric modifications.

Abstract

Starting with a trivial periodic flow on $\mathbb{S}M$, the unit tangent bundle of a genus two surface, we perform a Dehn-type surgery on the manifold around a tubular neighborhood of a curve on $\mathbb{S}M$ that projects to a self-intersecting closed geodesic on $M$, to get a surgered flow which restricted to the surgery region is ergodic with respect to the volume measure. The surgered flow projects to a map on the surgery track that can be taken to be a linked twist map with oppositely oriented shears which generates the ergodic behavior for sufficiently strong shears in the surgery.