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Partially hyperbolic geodesic flow via conformal deformation

Ygor de Jesus, Luis Pedro Piñeyrúa, Sergio Romaña

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.

Partially hyperbolic geodesic flow via conformal deformation

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.

Paper Structure

This paper contains 17 sections, 23 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemannian Geometry
  4. Geodesic Flow
  5. Partial Hyperbolicity
  6. Cone invariance criteria for Partially Hyperbolicity
  7. Breaking the Anosov Property
  8. The new geodesic flow is partially hyperbolic
  9. Cone invariance for the initial metric
  10. Cone invariance for the deformed metric
  11. Cone invariance for parallel geodesics
  12. Cone invariance for almost parallel geodesics
  13. Cone invariance for $s$-transversal geodesics
  14. Proof of Theorem \ref{['there exist conformal partially hyperbolic']}
  15. Proof of Theorem \ref{['ergodic partially hyperbolic geodesic flow']}
  16. ...and 2 more sections

Key Result

Theorem A

Let $(M,g)$ be a compact Kähler manifold of holomorphic curvature $-1$ or a compact locally symmetric quaternionic Kähler manifold of negative curvature. Then there exists a conformal metric $\widetilde{g}=\phi g$ such that the geodesic flow $\widetilde{g}_t$ is partially hyperbolic and not Anosov.

Figures (2)

  • Figure 1: Behavior of $r_2$
  • Figure 2: Graph of the function $p$ for $\varepsilon=1$

Theorems & Definitions (40)

  • Theorem A
  • Conjecture 1.1: KATOK1982ENTROPY
  • Theorem B
  • Corollary B.1
  • Corollary B.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • Definition 2.1
  • ...and 30 more