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Analytic symplectomorphisms displaying minimal ergodicity on the sphere, cylinder and disk

Yann Delaporte

Abstract

We construct analytic symplectomorphisms on the sphere, the disk and the cylinder which are minimally ergodic (only 3 ergodic measures). To achieve this, we apply and generalize a principle introduced by Berger, based on the Approximation by Conjugacy method of Anosov-Katok.

Analytic symplectomorphisms displaying minimal ergodicity on the sphere, cylinder and disk

Abstract

We construct analytic symplectomorphisms on the sphere, the disk and the cylinder which are minimally ergodic (only 3 ergodic measures). To achieve this, we apply and generalize a principle introduced by Berger, based on the Approximation by Conjugacy method of Anosov-Katok.
Paper Structure (29 sections, 38 theorems, 141 equations, 4 figures)

This paper contains 29 sections, 38 theorems, 141 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Surfaces and spaces
  3. Kantorovich-Wasserstein distance
  4. Schemes and Principles
  5. AbC Principle
  6. AbC$^\star$ Principle
  7. Minimal ergodicity
  8. Lemma for minimal ergodicity
  9. AbC$^\star$ scheme
  10. Realization of minimal ergodicity
  11. Proof of the AbC$^\star$ principle
  12. Approximation theorem
  13. Complexification and structures of $\mathbb{M}$
  14. Complexification of $\mathbb{M} \in \left\lbrace \mathbb{A} ,\mathbb{D}, \mathbb{S} \right\rbrace$
  15. Complexification of the cylinder $\mathbb{A}$.
  16. ...and 14 more sections

Key Result

Theorem A

There exist minimally ergodic analytic symplectomorphisms on the cylinder, the sphere and the disk.

Figures (4)

  • Figure 1: Bicurve on the cylinder
  • Figure 2: Lemma for minimal ergodicity
  • Figure 3: Symplectomorphism $\phi$ straightening the bicurve.
  • Figure :

Theorems & Definitions (65)

  • Theorem A
  • Definition 2.1: AbC scheme
  • Definition 2.2
  • Proposition 2.3: berger_analytic_2024
  • Theorem 2.4: AbC principle
  • Definition 2.5: Bicurve
  • Remark 2.6
  • Remark 2.8
  • Remark 2.9
  • Definition 2.10: AbC$^\star$ scheme
  • ...and 55 more