Partially hyperbolic geodesic flow via conformal deformation

TL;DR

This paper introduces a novel method using conformal deformations to construct non-Anosov partially hyperbolic geodesic flows, providing new examples with specific dynamical properties and analyzing their ergodic behavior.

Contribution

It presents a new conformal deformation technique for constructing partially hyperbolic geodesic flows, simplifying conditions and enabling the creation of diverse examples.

Findings

Constructed non-Anosov partially hyperbolic geodesic flows using conformal deformations.

Proved ergodicity of the Liouville measure for these flows.

Produced metrics with partially hyperbolic flows and conjugate points.

Abstract

This paper presents a new construction of non-Anosov Partially Hyperbolic Geodesic flows. Our construction is closely related to the construction made by Carneiro and Pujals, the novelty is the use of conformal deformations to produce the examples. Some of the necessary conditions appear more naturally and are easier to check. Besides that, we could enumerate the conditions required to produce partially hyperbolic geodesic flow examples. We show how to produce examples with metrics that are non-positively curved and with a finner analysis we can prove ergodicity for the Liouville measure and uniqueness of the measure of maximal entropy. These examples lie on the boundary of Anosov metrics, allowing us to also produce metrics with partially hyperbolic geodesic flows and conjugate points.