Iteration problem for distributional chaos

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.