P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy

Measure Expansivity and Specification for Pointwise Dynamics

“…In [3], it was proved that every homeomorphism on a compact metric space possessing specification property has positive topological entropy. In [1], authors have shown that P -chaotic map on a continuum has positive entropy. In [13], Moothathu has proved that continuous self-map on a compact metric space possessing shadowing property having either a non-minimal recurrent point or a sensitive minimal subsystem, has positive topological entropy.…”

Devaney chaos plus shadowing implies distributional chaos

“…Another natural question inspired by [1] is whether Devaney chaos under the shadowing property can imply distributional chaos. In this note we shall give a positive answer to this question.…”

“…Let (X , T ) be a non-periodic transitive system and has the shadowing property. Then (1) if (X , T ) has a fixed point, then there exists a dense Mycielski subset K of X such that K is distributionally n-δ n -scrambled for all n ≥ 2 and some δ n > 0. (2) if (X , T ) has a periodic point, then there exists a Mycielski subset K of X such that K is distributionally n-δ n -scrambled for all n ≥ 2 and some δ n > 0.…”

Shift spaces, ω-chaos and specification property

“…Recently, many papers were published on distributional chaos (e.g. see [1][2][3]13,[19][20][21]28]) however not too much is known about ω-chaos. Presently it is well know how to construct ω-scrambled sets for interval maps [16] (even very large sets of this kind [27]), however beyond dimension one only a few examples are known [7,12,15,23].…”

“…A set Ω ⊂ X containing at least two points is called an ω-scrambled set for f if, for any two x = y in Ω, (1) …”