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Iteration problem for distributional chaos (1606.08612v1)

Published 28 Jun 2016 in math.DS

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. $fn$ 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 $d(fk(x),fk(y))<\delta$ 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.

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