On some notions of chaos in dimension zero

By R. Pikuła. 2007

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Stable sets and mean Li–Yorke chaos in positive entropy systems (2012, background): “oduced in [28], and there are at least three versions of distributional chaos in the literature (DC1, DC2 and DC3), see [2] for the details. It is known that positive entropy does not imply DC1 chaos [27]. In [29] Sm´ıtal conjectured that positiveentropy impliesDC2 chaos, and Oprocha showedthat this”
DISTRIBUTIONAL CHAOS REVISITED (2009, background): “According to the author’s knowledge, there exist only two papers [58, 47] with examples of positive topological entropy systems with no distributionally chaotic pairs of type 1.”
Positive topological entropy implies chaos DC2 (2011, background): “hree variants of so-called distributional chaos DC1, DC2 and DC3 (ordered from strongest to weakest). Around 2000 Pikula showed that positive topological entropy does not imply DC1 (published in 2007 [P]). Sm´ıtal conjectured that it does imply DC2 (see e.g. [S2]). This problem has been around for s”
Distributional chaos for backward shifts (2009, background): “, there are systems with positive topological entropy which do not exhibit distributional chaos [45] and vice-versa [37]).”