Some remarks on dynamical system of solenoids. (English) Zbl 1405.37022

Biś, Andrzej; Kozłowski, Wojciech

Taiwanese J. Math. 22, No. 6, 1463-1478 (2018).

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 {M. Brin} and {A. Katok} [Lect. Notes Math. 1007, 30–38 (1983; Zbl 0533.58020)] for a single map, to a case of a solenoid. Lower estimations of these quantities by corresponding local measure entropies are described.

MSC:

37B40	Topological entropy
37B45	Continua theory in dynamics
28D20	Entropy and other invariants
54H20	Topological dynamics (MSC2010)
54F45	Dimension theory in general topology

Keywords:

entropy; local measure entropy; entropy-like quantity; solenoids; Carathéodory structure

Citations:

Zbl 0533.58020

References:

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