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An obstruction to fiberwise Anosov flows over 3-dimensional Anosov flows

Neige Paulet, Danyu Zhang

TL;DR

This work addresses which 3D Anosov flows can serve as bases for fiberwise Anosov flows, proving a non-existence obstruction when the base has infinitely many periodic orbits in a single free homotopy class. The main idea is to compare entropy growth along the fibers: for a base orbit of period $\tau$, the fiber’s first return map has entropy at least $K\tau$, but monodromies for freely homotopic base orbits yield conjugate maps with the same entropy, creating a contradiction as $\tau$ grows. Consequently, any ${\mathbb R}$-covered base flow that is not a suspension or a geodesic flow is precluded from being the base of a fiberwise Anosov flow, significantly restricting liftable 3D Anosov dynamics. The results illuminate rigidity constraints for constructing non-algebraic fiberwise Anosov flows and guide future search toward bases with finite free homotopy classes, such as the Bonatti–Langevin example, or totally periodic flows.

Abstract

We study obstructions preventing a three-dimensional Anosov flow from serving as the base of a fiberwise Anosov flow. We prove a non-existence result if the base flow admits infinitely many periodic orbits in the same free homotopy class. We get as a corollary that any R-covered Anosov flow serving as the base of a fiberwise Anosov flow is orbit equivalent to a suspension or a geodesic flow.

An obstruction to fiberwise Anosov flows over 3-dimensional Anosov flows

TL;DR

This work addresses which 3D Anosov flows can serve as bases for fiberwise Anosov flows, proving a non-existence obstruction when the base has infinitely many periodic orbits in a single free homotopy class. The main idea is to compare entropy growth along the fibers: for a base orbit of period , the fiber’s first return map has entropy at least , but monodromies for freely homotopic base orbits yield conjugate maps with the same entropy, creating a contradiction as grows. Consequently, any -covered base flow that is not a suspension or a geodesic flow is precluded from being the base of a fiberwise Anosov flow, significantly restricting liftable 3D Anosov dynamics. The results illuminate rigidity constraints for constructing non-algebraic fiberwise Anosov flows and guide future search toward bases with finite free homotopy classes, such as the Bonatti–Langevin example, or totally periodic flows.

Abstract

We study obstructions preventing a three-dimensional Anosov flow from serving as the base of a fiberwise Anosov flow. We prove a non-existence result if the base flow admits infinitely many periodic orbits in the same free homotopy class. We get as a corollary that any R-covered Anosov flow serving as the base of a fiberwise Anosov flow is orbit equivalent to a suspension or a geodesic flow.
Paper Structure (7 sections, 3 theorems, 18 equations)

This paper contains 7 sections, 3 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Acknowledgments
  4. Preliminaries
  5. Fiberwise Anosov flows
  6. Entropy
  7. Infinitely many freely homotopic periodic orbits

Key Result

Theorem 1

Let $\phi^t$ an Anosov flow on a 3-manifold $M$, such that there is infinitely many periodic orbits in the same free homotopy class. Then there is no fiberwise Anosov flow covering $\phi^t$.

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 1.1
  • Definition 2.1: Anosov flow
  • Definition 2.2: Fiberwise Anosov
  • Example 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem \ref{['thm: non existence of fiberwise if infinitely homotopic orbit']}
  • Remark 3.1
  • ...and 3 more