Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$C^1$-generic continuum-wise expansive surface diffeomorphisms

Alfonso Artigue, Bernardo Carvalho, José Cueto

Abstract

We exhibit a local residual set of surface $C^1$ diffeomorphisms that are continuum-wise expansive but are not expansive. We also exhibit an open and dense set of surface $C^1$ diffeomorphisms where expansiveness implies being Anosov.

$C^1$-generic continuum-wise expansive surface diffeomorphisms

Abstract

We exhibit a local residual set of surface diffeomorphisms that are continuum-wise expansive but are not expansive. We also exhibit an open and dense set of surface diffeomorphisms where expansiveness implies being Anosov.
Paper Structure (7 sections, 14 theorems, 77 equations, 8 figures)

This paper contains 7 sections, 14 theorems, 77 equations, 8 figures.

Table of Contents

  1. Introduction
  2. C1-generic expansive surface diffeomorphisms
  3. Generic cw-expansiveness
  4. Construction of DA on the Bitorus
  5. Cw-expansive diffeomorphisms
  6. Perturbation Lemmas
  7. Local Density of Cw-expansivity

Key Result

Theorem 1

Let $M$ be a closed surface and $\mathcal{A} \subset \mathrm{Diff}^1(M)$ be the set of Anosov diffeomorphisms. Then there exists an open and dense subset $\mathcal{B} \subset \mathrm{Diff}^1(M) \setminus \overline{\mathcal{A}}$ such that no diffeomorphism in $\mathcal{B}$ is expansive.

Figures (8)

  • Figure 1: In this picture $E$ denotes the set of expansive diffeomorphisms, $MS$ denotes the set of Morse-Smale diffeomorphisms, and $DA$ denotes the Derived-from-Anosov diffeomorphism. Also we show the local residual set ${\mathcal{R}}$ of Theorem \ref{['theorem:main_cw_exp1']}.
  • Figure 2: Foliations $W^s$ and $W^u$ in the annulus $\widetilde{A}.$
  • Figure 3: Bump function $\varphi$ and its derivative.
  • Figure 4: Diffeomorphism $h$.
  • Figure 5: Adding arbitrary curvature in the rectangle $R$.
  • ...and 3 more figures

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Definition 2.1: Local stable/unstable sets
  • Definition 2.2: Expansiveness
  • Definition 2.3: Periodic and non-wandering points
  • Definition 2.4: Sinks and Sources
  • Definition 2.5: Axiom A diffeomorphisms
  • Definition 2.6: No-cycle condition
  • Definition 2.7: Quasi-Anosov diffeomorphism
  • Definition 2.8: Quasi-transversality condition
  • ...and 29 more