Specification property and distributional chaos almost everywhere

Our main result shows that a continuous map f acting on a compact metric space (X,\rho ) with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set S dense in X which i...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2008-11, Vol.136 (11), p.3931-3940
Main Authors: OPROCHA, Piotr, STEFANKOVA, Marta
Format: Article
Language:English
Subjects:
Cantor set
Chaos theory
Exact sciences and technology
General mathematics
General topology
General, history and biography
Hausdorff dimensions
Homeomorphism
Integers
Lebesgue measures
Mathematical functions
Mathematical intervals
Mathematics
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:Our main result shows that a continuous map f acting on a compact metric space (X,\rho ) with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set S dense in X which is the union of disjoint sets homeomorphic to Cantor sets so that, for any two distinct points u,v\in S, the upper distribution function is identically 1 and the lower distribution function is zero at some \varepsilon >0. As a consequence, we describe a class of maps with a scrambled set of full Lebesgue measure in the case when X is the k-dimensional cube I^{k}. If X=I, then we can even construct scrambled sets whose complements have zero Hausdorff dimension.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-08-09602-0