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Weak mixing for area preserving flows on surfaces

Adam Kanigowski, Alexey Okunev, Rigoberto Zelada

Abstract

Let $(φ_t)$ be an area-preserving smooth flow on a compact, connected, orientable surface $\mathcal M$ with at least one but finitely many fixed points. Assume that $(φ_t)$ is analytic (up to a canonical change of coordinates) in the neighborhood of each saddle fixed point. We show that the flow $(φ_t)$ is weakly mixing on each of its (finitely many) quasi-minimal components.

Weak mixing for area preserving flows on surfaces

Abstract

Let be an area-preserving smooth flow on a compact, connected, orientable surface with at least one but finitely many fixed points. Assume that is analytic (up to a canonical change of coordinates) in the neighborhood of each saddle fixed point. We show that the flow is weakly mixing on each of its (finitely many) quasi-minimal components.
Paper Structure (23 sections, 26 theorems, 63 equations, 2 figures)

This paper contains 23 sections, 26 theorems, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the proof of Theorem A.
  3. Background on interval exchange transformations
  4. Induced IETs and their discontinuities
  5. A sufficient condition for minimality
  6. Background on smooth area preserving flows
  7. Special flow representations of smooth area-preserving flows
  8. The orthogonal flow
  9. First return maps are IETs
  10. Sketch of the proof of Theorem \ref{['t:decomposition']}
  11. Proof of Theorem \ref{['thm:AnalyticFlowRepresentation']}
  12. Analytic singularities of area-preserving flows
  13. Results
  14. Resolution of singularities and its corollaries
  15. Roof function near its discontinuities
  16. ...and 8 more sections

Key Result

Proposition 2.1

Let $N>1$, let $T:[0,1)\rightarrow[0,1)$ be a right-continuous IET with $N-1$ discontinuities, let $\Delta=[a,b)\subseteq [0,1)$, and let $T_\Delta$ be the map induced by $T$ on $\Delta$. If $\alpha\in \chi_\Delta$ is a discontinuity, then (a) $T^i\alpha\not\in [a,b)$ for $i\in\{1,...,h_{\Delta,T}(\

Figures (2)

  • Figure 1: Homoclinic loop of a locally Hamiltonian flow. Figure taken from FKZ.
  • Figure 2: An example of a special flow satisfying the conclusions of Theorem \ref{['thm:AnalyticFlowRepresentation']}.

Theorems & Definitions (54)

  • Remark 1.1
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3: Keane's Condition, one-sided variant
  • Theorem 3.1: MaiKat73NZ
  • Remark 3.2
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • ...and 44 more