Chaos among self-maps of the Cantor space

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:381

Article

D'Aniello, Emma¹  Darji, Udayan B.² 

[1] Univ Naples 2, Dipartimento Matemat, I-81100 Caserta, Italy

[2] Univ Louisville, Dept Math, Louisville, KY 40208 USA

关键词: Devaney chaos;    Entropy;    Generic map;   

DOI  :  10.1016/j.jmaa.2011.03.065

来源: Elsevier

【 摘 要 】

Glasner and Weiss have shown that a generic homeomorphism of the Cantor space has zero topological entropy. Hochman has shown that a generic transitive homeomorphism of the Cantor space has the property that it is topologically conjugate to the universal odometer and hence far from being chaotic in any sense. We show that a generic selfmap of the Cantor space has zero topological entropy. Moreover, we show that a generic self-map of the Cantor space has no periodic points and hence is not Devaney chaotic nor Devaney chaotic on any subsystem. However, we exhibit a dense subset of the space of all self-maps of the Cantor space each element of which has infinite topological entropy and is Devaney chaotic on some subsystem. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free