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Topological entropy and orbit growth in link complements

Matthias Meiwes

Abstract

In this article, we exhibit certain linking properties of periodic orbits of $C^{1+α}$ flows with positive topological entropy on closed 3-manifolds M. It is shown that any such flow contains a link L of periodic orbits and a horseshoe K in MŁ, such that all periodic orbits in K are unique in their homotopy class in MŁ(among periodic orbits in M). Moreover, the entropy of the flow can be approximated by the entropies of such horseshoes K. A version of that result for chords is obtained. Our main motivation comes from Reeb dynamics, and as an application, we address a question by Alves-Pirnapasov, and obtain that the topological entropy of a 3-dimensional, $C^{\infty}$-generic Reeb flow can be approximated by the exponential homotopical growth rates of contact homology in link complements.

Topological entropy and orbit growth in link complements

Abstract

In this article, we exhibit certain linking properties of periodic orbits of flows with positive topological entropy on closed 3-manifolds M. It is shown that any such flow contains a link L of periodic orbits and a horseshoe K in MŁ, such that all periodic orbits in K are unique in their homotopy class in MŁ(among periodic orbits in M). Moreover, the entropy of the flow can be approximated by the entropies of such horseshoes K. A version of that result for chords is obtained. Our main motivation comes from Reeb dynamics, and as an application, we address a question by Alves-Pirnapasov, and obtain that the topological entropy of a 3-dimensional, -generic Reeb flow can be approximated by the exponential homotopical growth rates of contact homology in link complements.
Paper Structure (20 sections, 24 theorems, 148 equations, 1 figure)

This paper contains 20 sections, 24 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Growth in the complement of a link
  3. An application to Reeb flows and forcing
  4. Horseshoes
  5. Preliminaries: The Markov flows of Lima-Sarig
  6. Main properties of the Markov flows of Lima-Sarig
  7. Adapted Poincaré sections
  8. Non-uniform hyperbolic set and Pesin charts
  9. Generalized pseudo orbits, admissible manifolds, and shadowing
  10. The Markov partition
  11. Construction of the horseshoes
  12. An approximation result for countable Markov shifts
  13. Separation of orbits
  14. The horseshoe
  15. The first horseshoe
  16. ...and 5 more sections

Key Result

Corollary 1.3

Let $(M,\xi)$ be a closed contact $3$-manifold. Then, for a $C^{\infty}$-open and dense set of supporting contact forms $\alpha$ the following holds. If $h_{\mathrm{top}}(\alpha)>0$, then there is a sequence of links $\mathcal{L}_k\subset \mathrm{Per}(\varphi_{\alpha})$ such that $\Gamma_{\mathcal{L

Figures (1)

  • Figure 1: Schematic picture of $\mathcal{Q}$, $D_0$, and one region $D_i$ with its image under $f^N$. Here, we assume that $\Theta_- \prec_s \Omega_i \prec_s \Theta_+$ and $\Theta_- \prec_u \Omega_i \prec \Theta_+$, and that $f^N$ keeps the orientation of the stable and unstable manifolds of all of the loops involved. The two dashed rectangles are $\mathcal{Q}(e_-^i; e_+^i)$ and $\mathcal{Q}(e_{-+}^i;e_{+-}^i)$. The coordinates are those of the Pesin chart, and note that in this picture, $s$-admissible manifolds are vertical and $u$-admissible manifolds horizontal.

Theorems & Definitions (55)

  • Definition 1.1: AlvesPirnapasov
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • proof : Proof of Corollary \ref{['thm:approximation_transverse']}
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • ...and 45 more