Entropy dimension of topological dynamical systems

Dou, Dou; Huang, Wen; Park, Kyewon

doi:10.1090/S0002-9947-2010-04906-2

Entropy dimension of topological dynamical systems
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by Dou Dou, Wen Huang and Kyewon Koh Park

Trans. Amer. Math. Soc. 363 (2011), 659-680

DOI: https://doi.org/10.1090/S0002-9947-2010-04906-2

Published electronically: September 2, 2010

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Abstract:

We introduce the notion of topological entropy dimension to measure the complexity of entropy zero systems. It measures the superpolynomial, but subexponential, growth rate of orbits. We also introduce the dimension set, $\mathcal {D}(X,T)\subset [0,1]$, of a topological dynamical system to study the complexity of its factors. We construct a minimal example whose dimension set consists of one number. This implies the property that every nontrivial open cover has the same entropy dimension. This notion for zero entropy systems corresponds to the $K$-mixing property in measurable dynamics and to the uniformly positive entropy in topological dynamics for positive entropy systems. Using the entropy dimension, we are able to discuss the disjointness between the entropy zero systems. Properties of entropy generating sequences and their dimensions have been investigated.

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