DISTRIBUTIONAL CHAOS REVISITED

By Piotr Oprocha. 2009

89 citations recorded by the source corpus.

In their famous paper, Schweizer and Smital introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval. Further, their approach was extended to the general compact metric space case. In this article we provide an example which shows that the definition of distributional chaos (and as a result Li-Yorke chaos) may be fulfilled by a dynamical system with (intuitively) regular dynamics embedded in ℝ 3 . Next, we state strengthened versions of distributional chaos which, as we show, are present in systems commonly considered to have complex dynamics. We also prove that any interval map with positive topological entropy contains two invariant subsets X,Y C I such that f|x has positive topological entropy and f|y displays distributional chaos of type 1, but not conversely.

Subfields: Economics and Econometrics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8a13738030f97512a6fdf982e9ee3d074be5745a

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Citations in the corpus

Li–Yorke and distributionally chaotic operators (2010, unknown): “s deﬁned by Fn xy(τ) = 1 n card{0 ≤ i ≤ n − 1 : d(fix, fiy) < τ} where card{A} denotes the cardinality of the set A. Deﬁne Fxy(τ) = liminf n→∞ Fn xy(τ) F∗ xy(τ) = limsup n→∞ Fn xy(τ). Deﬁnition 2 ([28, 26]). A continuous map f : X → X on a metric space X is distributionally chaotic if there exist”
Positive topological entropy implies chaos DC2 (2011, background): “(i.e., the number t 0 and the lower density δ) are common for all pairs (which is good, in the sense that the chaos is strong). Such situation is called uniform DC2 (see [BS], see also uniform DC1 in [O2]). On the other hand, the scrambling is “synchronic”, i.e., all points in the scrambled set get ”
Distributional chaos for operators with full scrambled sets (2013, background): “Following [14], we will consider only the definition of uniform distributional chaos, which is one of the strongest possibilities (see [16]).”
Distributional chaos for linear operators (2013, background): “The following notions were introduced in [23] and [21], respectively.”