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Homoclinic tangency of codimension 2 and historic wandering domains

Kodai Yamamoto

Abstract

We construct a four-dimensional diffeomorphism exhibiting a homoclinic tangency of the largest codimension, which admits a historic wandering domain of positive Lebesgue measure. Every orbit in this wandering domain exhibits historic behavior, in the sense that time averages do not converge. This example shows that homoclinic tangencies of the largest codimension can still give rise to positive Lebesgue measure sets with non-convergent statistical behavior.

Homoclinic tangency of codimension 2 and historic wandering domains

Abstract

We construct a four-dimensional diffeomorphism exhibiting a homoclinic tangency of the largest codimension, which admits a historic wandering domain of positive Lebesgue measure. Every orbit in this wandering domain exhibits historic behavior, in the sense that time averages do not converge. This example shows that homoclinic tangencies of the largest codimension can still give rise to positive Lebesgue measure sets with non-convergent statistical behavior.
Paper Structure (16 sections, 13 theorems, 186 equations, 6 figures)

This paper contains 16 sections, 13 theorems, 186 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic set and homoclinic points
  4. Statistical properties
  5. Related works
  6. Plan of this paper
  7. Construction of the Map
  8. Return to the Homoclinic Tangency
  9. Unstable Boxes and Stable Boxes
  10. Linked rectangles
  11. Proof of Theorem \ref{['main2']}
  12. Proof of Main Theorem
  13. Perturbation
  14. Code conditions for wandering domains
  15. Construction of wandering domains
  16. ...and 1 more sections

Key Result

Theorem 1.1

There exist sequences of diffeomorphisms $F_{n}$ and $G_{n}$ converging to $F$ in the $C^{r}$-topology $(1 \leq r < \infty)$ such that the following hold:

Figures (6)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4: The perturbed linked pairs $(\delta_{1} + \widetilde{B}^{s}_{1}, \widetilde{B}^{u}_{1})$ and $(\delta_{1} + B^{s}_{1},B^{u}_{1})$.
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.6
  • ...and 18 more