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July, 2020 Finite-to-one zero-dimensional covers of dynamical systems
Hisao KATO, Masahiro MATSUMOTO
J. Math. Soc. Japan 72(3): 819-845 (July, 2020). DOI: 10.2969/jmsj/82128212


In this paper, we study the existence of finite-to-one zero-dimensional covers of dynamical systems. Kulesza showed that any homeomorphism $f:X \to X$ on an $n$-dimensional compactum $X$ with zero-dimensional set $P(f)$ of periodic points can be covered by a homeomorphism on a zero-dimensional compactum via an at most $(n + 1)^n$-to-one map. Moreover, Ikegami, Kato and Ueda showed that in the theorem of Kulesza, the condition of at most $(n + 1)^n$-to-one map can be strengthened to the condition of at most $2^n$-to-one map. In this paper, we will show that the theorem is also true for more general maps except for homeomorphisms. In fact we prove that the theorem is true for a class of maps containing two-sided zero-dimensional maps. For the special case, we give a theorem of symbolic extensions of positively expansive maps. Finally, we study some dynamical zero-dimensional decomposition theorems of spaces related to such maps.


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Hisao KATO. Masahiro MATSUMOTO. "Finite-to-one zero-dimensional covers of dynamical systems." J. Math. Soc. Japan 72 (3) 819 - 845, July, 2020.


Received: 5 February 2019; Published: July, 2020
First available in Project Euclid: 28 February 2020

zbMATH: 07257212
MathSciNet: MR4125847
Digital Object Identifier: 10.2969/jmsj/82128212

Primary: 37B10
Secondary: 37B45 , 37C45 , 54F45 , 54H20‎

Keywords: dimension , dynamical systems , finite-to-one map , general position , periodic point , positively expansive map , zero-dimensional covers (extensions) of dynamical systems

Rights: Copyright © 2020 Mathematical Society of Japan


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Vol.72 • No. 3 • July, 2020
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