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Invariant sets for homeomorphisms of hyperbolic 3-manifolds

Elena Gomes, Santiago Martinchich, Rafael Potrie

TL;DR

The paper analyzes invariant sets for homeomorphisms of closed hyperbolic 3-manifolds homotopic to the identity, introducing escape-rate and homological-rotation frameworks. It shows that minimality is incompatible with uniform positive escape rates and that quasi-geodesic dynamics yield uncountably many disjoint invariant sets with positive entropy, via shadowing by geodesic flows and a regulating pseudo-Anosov flow when foliations are uniform and $ ext{R}$-covered. The authors develop constructions of invariant sets $ ilde ext{T}_eta$ near regular periodic orbits and prove a dichotomy: either zero asymptotic displacement on a dense set or global quasi-geodesic behavior, with implications for finite covers and homological directions. Extending to foliations, they demonstrate that positive escape rate induces a rich invariant-set structure and high entropy, and they discuss homological-rotation consequences, finite lifts, and extensions to non-hyperbolic manifolds. The results deepen the understanding of minimality, shadowing, and the geometry of invariant sets in 3-manifold dynamics, and raise open questions about broader manifold classes and flows.

Abstract

We prove that under some assumptions on how points escape to infinity in the universal cover, homeomorphisms of hyperbolic 3-manifolds are forced to have several invariant sets (in particular, they cannot be minimal). For this, we use some shadowing techniques which, when the homeomorphism has positive speed with respect to a uniform foliation, allow us to obtain strong consequences on the structure of the invariant sets. We discuss also homological rotation sets and end the paper with some extensions to other manifolds as well as posing some general problems for the understanding of minimal homeomorphisms of 3-manifolds.

Invariant sets for homeomorphisms of hyperbolic 3-manifolds

TL;DR

The paper analyzes invariant sets for homeomorphisms of closed hyperbolic 3-manifolds homotopic to the identity, introducing escape-rate and homological-rotation frameworks. It shows that minimality is incompatible with uniform positive escape rates and that quasi-geodesic dynamics yield uncountably many disjoint invariant sets with positive entropy, via shadowing by geodesic flows and a regulating pseudo-Anosov flow when foliations are uniform and -covered. The authors develop constructions of invariant sets near regular periodic orbits and prove a dichotomy: either zero asymptotic displacement on a dense set or global quasi-geodesic behavior, with implications for finite covers and homological directions. Extending to foliations, they demonstrate that positive escape rate induces a rich invariant-set structure and high entropy, and they discuss homological-rotation consequences, finite lifts, and extensions to non-hyperbolic manifolds. The results deepen the understanding of minimality, shadowing, and the geometry of invariant sets in 3-manifold dynamics, and raise open questions about broader manifold classes and flows.

Abstract

We prove that under some assumptions on how points escape to infinity in the universal cover, homeomorphisms of hyperbolic 3-manifolds are forced to have several invariant sets (in particular, they cannot be minimal). For this, we use some shadowing techniques which, when the homeomorphism has positive speed with respect to a uniform foliation, allow us to obtain strong consequences on the structure of the invariant sets. We discuss also homological rotation sets and end the paper with some extensions to other manifolds as well as posing some general problems for the understanding of minimal homeomorphisms of 3-manifolds.

Paper Structure

This paper contains 29 sections, 52 theorems, 28 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Precise statement of results and preliminaries
  3. Escape rate
  4. Homological rotation
  5. The quasi-geodesic case
  6. $\mathbb R$-covered foliations
  7. Quasi-geodesic homeomorphisms
  8. Constructing the invariant set
  9. Invariant subsets
  10. Topological entropy
  11. Proof of Theorem \ref{['teoB']}
  12. Positive escape rate against foliations
  13. The quasi-geodesic property
  14. Some properties of the pseudo-Anosov regulating flow
  15. Contraction and expansion of lines
  16. ...and 14 more sections

Key Result

Theorem A

Let $f: M \to M$ be a quasi-geodesic homeomorphism of a closed hyperbolic 3-manifold. Then, $f$ contains infinitely many disjoint compact invariant sets and has positive topological entropy.

Figures (6)

  • Figure 1: Ilustration of Proposition \ref{['prop exists T sub gamma']}.
  • Figure 2: The figure depicts a deformation of the flow in a neighborhood fully consisting of periodic orbits of same period. After the deformation, in the middle section one gets orbits going in both directions.
  • Figure 3: A regular orbit of a pseudo-Anosov flow on the left, and a 3-pronged singular orbit on the right.
  • Figure 4: A stable line through a $3$-prong.
  • Figure 5: $U^r$ and $V^r$ viewed on a leaf $L$.
  • ...and 1 more figures

Theorems & Definitions (103)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.6
  • Proposition 2.7
  • Theorem 2.8
  • ...and 93 more