Iteration problem for distributional chaos

TL;DR

The paper disproves the iteration invariance of distributional chaos type 3 and introduces a new, stronger form called DC2½ that is preserved under iteration, conjugacy, and implies Li-Yorke chaos, with zero topological entropy.

Contribution

It shows that DC3 chaos is not iteration invariant and proposes DC2½ as a new, stronger form of distributional chaos that is preserved under iteration and conjugacy.

Findings

DC3 chaos is not iteration invariant.

DC2½ chaos is preserved under iteration.

DC2½ chaos implies Li-Yorke chaos and has zero topological entropy.

Abstract

We disprove the conjecture that distributional chaos of type 3 (briefly, DC3) is iteration invariant and show that a slightly strengthened definition, denoted by DC2$\frac{1}{2}$, is preserved under iteration, i.e. $f^n$ is DC2$\frac{1}{2}$ if and only if $f$ is too. Unlike DC3, DC2$\frac{1}{2}$ is also conjugacy invariant and implies Li-Yorke chaos. The definition of DC2$\frac{1}{2}$ is the following: a pair $(x,y)$ is DC2$\frac{1}{2}$ iff $\Phi_{(x,y)}(0)<\Phi^*_{(x,y)}(0)$, where $\Phi_{(x,y)}(\delta)$ (resp. $\Phi^*_{(x,y)}(\delta)$) is lower (resp. upper) density of times $k$ when $d(f^k(x),f^k(y))<\delta$ and both densities are defined at 0 as limits of their values for $\delta\to 0^+$. Hence DC$2\frac{1}{2}$ shares similar properties with DC1 and DC2 but unlike them, strict DC$2\frac{1}{2}$ systems must have zero topological entropy.