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Relativization of sensitivity in minimal systems

  • TAO YU (a1) and XIAOMIN ZHOU (a2)

Abstract

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 $\unicode[STIX]{x1D70B}$ .

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