Some Remarks on Dynamical System of Solenoids

We show that a solenoid is a dynamical object and we express its complexity by a number of different entropy-like quantities in Hurley’s sense. Some relations between these entropy-like quantities are presented. We adopt the theory of Carathéodory dimension structures introduced axiomatically by Pes...

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Bibliographic Details
Published in:Taiwanese journal of mathematics 2018-12, Vol.22 (6), p.1463-1478
Main Authors: Biś, Andrzej, Kozłowski, Wojciech
Format: Article
Language:English
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Summary:We show that a solenoid is a dynamical object and we express its complexity by a number of different entropy-like quantities in Hurley’s sense. Some relations between these entropy-like quantities are presented. We adopt the theory of Carathéodory dimension structures introduced axiomatically by Pesin to a case of a solenoid. Any of the above mentioned entropy-like quantities determines a particular Carathéodory structure such that its upper capacity coincides with the considered quantity. We mimic a definition of the local measure entropy, introduced by Brin and Katok for a single map, to a case of a solenoid. Lower estimations of these quantities by corresponding local measure entropies are described.
ISSN:1027-5487
2224-6851
DOI:10.11650/tjm/180506