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Relativization of sensitivity in minimal systems

Part of:Topological dynamicsConnections with other structures, applicationsErgodic theory

Published online by Cambridge University Press:  27 December 2018

TAO YU and
XIAOMIN ZHOUOpen the ORCID record for XIAOMIN ZHOU[Opens in a new window]
TAO YU
Affiliation:
Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, PR China email ytnuo@mail.ustc.edu.cn
XIAOMIN ZHOU
Affiliation:
Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, PR China email zxm12@mail.ustc.edu.cn
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Abstract

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Let $\unicode[STIX]{x1D70B}:(X,T)\rightarrow (Y,S)$ be an extension between minimal systems; we consider its relative sensitivity. We obtain that $\unicode[STIX]{x1D70B}$ is relatively $n$-sensitive if and only if the relative $n$-regionally proximal relation contains a point whose coordinates are distinct; and the structure of $\unicode[STIX]{x1D70B}$ which is relatively $n$-sensitive but not relatively $(n+1)$-sensitive is determined. Let ${\mathcal{F}}_{t}$ be the families consisting of thick sets. We introduce notions of relative block ${\mathcal{F}}_{t}$-sensitivity and relatively strong ${\mathcal{F}}_{t}$-sensitivity. Let $\unicode[STIX]{x1D70B}:(X,T)\rightarrow (Y,S)$ be an extension between minimal systems. Then the following Auslander–Yorke type dichotomy theorems are obtained: (1) $\unicode[STIX]{x1D70B}$ is either relatively block ${\mathcal{F}}_{t}$-sensitive or $\unicode[STIX]{x1D719}:(X,T)\rightarrow (X_{\text{eq}}^{\unicode[STIX]{x1D70B}},T_{\text{eq}})$ is a proximal extension where $(X_{\text{eq}}^{\unicode[STIX]{x1D70B}},T_{\text{eq}})\rightarrow (Y,S)$ is the maximal equicontinuous factor of $\unicode[STIX]{x1D70B}$. (2) $\unicode[STIX]{x1D70B}$ is either relatively strongly ${\mathcal{F}}_{t}$-sensitive or $\unicode[STIX]{x1D719}:(X,T)\rightarrow (X_{d}^{\unicode[STIX]{x1D70B}},T_{d})$ is a proximal extension where $(X_{d}^{\unicode[STIX]{x1D70B}},T_{d})\rightarrow (Y,S)$ is the maximal distal factor of $\unicode[STIX]{x1D70B}$.

Keywords

relative sensitivityminimalityequicontinuous factordistal factor

MSC classification

Primary:37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary:37A05: Measure-preserving transformations54H20: Topological dynamics

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020, pp. 1715 - 1728
Copyright
© Cambridge University Press, 2018

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