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Sufficient conditions for distributional chaos of type I

Noriaki Kawaguchi

TL;DR

The paper develops simple, verifiable sufficient conditions for distributional chaos of type I (DC1) in topological dynamics, motivated by the time-one map of a mixing Anosov flow. It introduces DC1_n and delta_n notions and shows that under a closed invariant subset Lambda with f|Lambda minimal and equicontinuous, the union of strong stable sets densely covers X and the product map f|Lambda × f is transitive, yielding residual DC1_n^{delta_n} for all 2 <= n < |Lambda|+1; Mycielski's theorem then provides DC1_n. The main contribution is Theorem 1.1, complemented by Furstenberg-family corollaries that extend DC1 conclusions to broader settings, including when Y = R(g,F) and f is Delta(F)^*-transitive. The proofs in Section 2 establish recurrence and product-transitivity lemmas (Lemma 1.2 and Lemma 1.3) that underpin the DC1 residual structure, thereby linking equicontinuous minimal dynamics, stable-set density, and transitivity to distributional chaos.

Abstract

Distributional chaos of type I (DC1) is a stronger variant of Li-Yorke chaos. In this paper, we consider the fact that the time-one map of a mixing Anosov flow exhibits DC1 and generalize it to obtain simple sufficient conditions for DC1.

Sufficient conditions for distributional chaos of type I

TL;DR

The paper develops simple, verifiable sufficient conditions for distributional chaos of type I (DC1) in topological dynamics, motivated by the time-one map of a mixing Anosov flow. It introduces DC1_n and delta_n notions and shows that under a closed invariant subset Lambda with f|Lambda minimal and equicontinuous, the union of strong stable sets densely covers X and the product map f|Lambda × f is transitive, yielding residual DC1_n^{delta_n} for all 2 <= n < |Lambda|+1; Mycielski's theorem then provides DC1_n. The main contribution is Theorem 1.1, complemented by Furstenberg-family corollaries that extend DC1 conclusions to broader settings, including when Y = R(g,F) and f is Delta(F)^*-transitive. The proofs in Section 2 establish recurrence and product-transitivity lemmas (Lemma 1.2 and Lemma 1.3) that underpin the DC1 residual structure, thereby linking equicontinuous minimal dynamics, stable-set density, and transitivity to distributional chaos.

Abstract

Distributional chaos of type I (DC1) is a stronger variant of Li-Yorke chaos. In this paper, we consider the fact that the time-one map of a mixing Anosov flow exhibits DC1 and generalize it to obtain simple sufficient conditions for DC1.

Paper Structure

This paper contains 2 sections, 12 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorems 1.1, Lemmas 1.2 and 1.3

Key Result

Theorem 1

Let $Z$ be a perfect complete separable metric space. If $R_n$ is a residual subset of $Z^n$ for each $n\ge2$, then there is a Mycielski set $S$ which is dense in $Z$ and satisfies $(x_1,x_2,\dots,x_n)\in R_n$ for all $n\ge2$ and distinct $x_1,x_2,\dots,x_n\in S$.

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem : Mycielski
  • Remark 1.1
  • Claim
  • Remark 1.2
  • Definition 1.2
  • Remark 1.3
  • Lemma 1.1
  • proof
  • Remark 1.4
  • ...and 23 more