Taiwanese Journal of Mathematics

Some Remarks on Dynamical System of Solenoids

Andrzej Biś and Wojciech Kozłowski

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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.

Article information

Taiwanese J. Math., Volume 22, Number 6 (2018), 1463-1478.

Received: 3 August 2017
Revised: 4 May 2018
Accepted: 20 May 2018
First available in Project Euclid: 9 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37B45: Continua theory in dynamics 28D20: Entropy and other invariants
Secondary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 54F45: Dimension theory [See also 55M10]

entropy local measure entropy entropy-like quantity solenoids Carathéodory structure


Biś, Andrzej; Kozłowski, Wojciech. Some Remarks on Dynamical System of Solenoids. Taiwanese J. Math. 22 (2018), no. 6, 1463--1478. doi:10.11650/tjm/180506. https://projecteuclid.org/euclid.twjm/1528509852

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