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arXiv · 1412.6928

On the weakest version of distributional chaos

Jana Hantáková·Samuel Roth·Zuzana Roth
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Abstract

The aim of the paper is to correct and improve some results concerning distributional chaos of type 3. We show that in a general compact metric space, distributional chaos of type 3, denoted DC3, even when assuming the existence of an uncountable scrambled set, is a very weak form of chaos. In particular, (i) the chaos can be unstable (it can be destroyed by conjugacy), and (ii) such an unstable system may contain no Li-Yorke pair. However, the definition can be strengthened to get DC$2\frac{1}{2}$ which is a topological invariant and implies Li-Yorke chaos, similarly as types DC1 and DC2; but unlike them, strict DC$2\frac{1}{2}$ systems must have zero topological entropy.

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Jana Hantáková, Samuel Roth, Zuzana Roth. 2014-12-22. On the weakest version of distributional chaos. https://arxiv.org/abs/1412.6928

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