Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems

Šotola, Jakub

doi:10.3934/dcds.2018225

Cited by 3 publications

(1 citation statement)

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“…Finally, it will be proved that (M(I), f ) has infinite topological entropy for any transitive autonomous system (I, f ), while there exists a transitive non-autonomous system (I, f 0,∞ ) such that (M(I), f ) has zero topological entropy. To proceed, we recall the definition of topological (sequence) entropy of (X, f 0,∞ ) introduced in [9,22], which will be discussed in detail in Section 5. Suppose that…”

Section: Transitivity and Mixingmentioning

confidence: 99%

Dynamics of continuous maps induced on the space of probability measures

Shao,

Zhu,

Chen

2019

Preprint

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Let f be a continuous self-map on a compact interval I and f be the induced map on the space M(I) of probability measures. We obtain a sharp condition to guarantee that (I, f ) is transitive if and only if (M(I), f ) is transitive. We also show that the sensitivity of (I, f ) is equivalent to that of (M(I), f ). We prove that (M(I), f ) must have infinite topological entropy for any transitive system (I, f ), while there exists a transitive nonautonomous system (I, f0,∞) such that (M(I), f0,∞) has zero topological entropy, where f0,∞ = {fn} ∞ n=0 is a sequence of continuous self-maps on I. For a continuous self-map f on a general compact metric space X, we show that chain transitivity of (X, f ) implies chain mixing of (M(X), f ), and we provide two counterexamples to demonstrate that the converse is not true. We confirm that shadowing of (X, f ) is not inherited by (M(X), f ) in general. For a non-autonomous system (X, f0,∞), we prove that Li-Yorke chaos (resp., distributional chaos) of (X, f0,∞) carries over to (M(X), f0,∞), and give an example to show that the converse may not be true. We prove that if fn is surjective for all n ≥ 0, then chain mixing of (M(X), f0,∞) always holds true, and shadowing of (M(X), f0,∞) implies topological mixing of (X, f0,∞). In addition, we prove that topological mixing (resp., mild mixing and topological exactness) of (X, f0,∞) is equivalent to that of (M(X), f0,∞), and that (X, f0,∞) is cofinitely sensitive if and only if (M(X), f0,∞) is cofinitely sensitive.

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Section: Transitivity and Mixingmentioning

confidence: 99%