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Mathers regions of instability for annulus diffeomorphisms

Salvador Addas-Zanata, Fabio Armando Tal

Abstract

Let $f$ be a $C^{1+\varepsilon}$ diffeomorphism of the closed annulus $A$ that preserves orientation and the boundary components, and $\widetilde{f}$ be a lift of $f$ to its universal covering space. Assume that $A$ is a Birkhoff region of instability for $f$, and the rotation set of $\widetilde{f}$ is a non-degenerate interval. Then there exists an open $f$-invariant annulus $A^*$ whose boundary intersects both boundary components of of $A$, and points $z^+$ and $z^-$ in $A^*$, such that the positive (resp. negative) orbit of $z^+$ converges to a set contained in the upper (resp. lower) boundary component of $A^*$ and the positive (resp. negative) orbit of $z^-$ converges to a set contained in the lower (resp. upper) boundary component of $A^*$. This extends a celebrated result originally proved by Mather for area-preserving twist diffeomorphisms.

Mathers regions of instability for annulus diffeomorphisms

Abstract

Let be a diffeomorphism of the closed annulus that preserves orientation and the boundary components, and be a lift of to its universal covering space. Assume that is a Birkhoff region of instability for , and the rotation set of is a non-degenerate interval. Then there exists an open -invariant annulus whose boundary intersects both boundary components of of , and points and in , such that the positive (resp. negative) orbit of converges to a set contained in the upper (resp. lower) boundary component of and the positive (resp. negative) orbit of converges to a set contained in the lower (resp. upper) boundary component of . This extends a celebrated result originally proved by Mather for area-preserving twist diffeomorphisms.
Paper Structure (12 sections, 17 theorems, 18 equations, 1 figure)

This paper contains 12 sections, 17 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction and statements of the results
  2. Preliminaries
  3. Prime ends compactification of open disks
  4. Topologically transverse intersections
  5. Some Pesin theory
  6. Some forcing results
  7. On maximal invariant sets
  8. Proofs
  9. Proof of Theorem 2
  10. Proof of Lemma 1
  11. Proof of Theorem 1
  12. Proof of Theorem 3

Key Result

Theorem 1

Let $f\in \rm{Diff}_0^{1+\varepsilon}(A)$ for some $\varepsilon>0$ be such that $A$ is a Birkhoff region of instability and for some fixed lift $\widetilde{f},$$\rho (\widetilde{f})$ has interior. Then there exists a homotopically non-trivial simple closed curve $\gamma \subset A$ and an $f$-invaria

Figures (1)

  • Figure 1: Sketch of the smooth diffeomorphism for which $A$ (obtained by gluing the two lateral sides together) is not a Mather Region of Instability.

Theorems & Definitions (17)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Proposition 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 7 more