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Measurable sensitivity via Furstenberg families

  • * Corresponding author: Tao Yu

    * Corresponding author: Tao Yu
The author was supported by NNSF of China (11371339,11431012,11571335).
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  • Let $(X, T)$ be a topological dynamical system, and $\mu$ be a $T$-invariant Borel probability measure on $X$. Let $\mathcal{F}$ be a family of subsets of $\mathbb{Z}_+$. We introduce notions of $\mathcal{F}$-sensitivity for $\mu$ and block $\mathcal{F}$-sensitivity for $\mu$.

    Let $\mathcal{F}_t$ (resp. $\mathcal{F}_{ip}$) be the families consisting of thick sets (resp. IP-sets). The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either $\mathcal{F}_{t}$-sensitive for $\mu$ or an almost one-to-one extension of its maximal equicontinous factor. (2) a minimal system is either block $\mathcal{F}_{t}$-sensitive for $\mu$ or a proximal extension of its maximal equicontinous factor. (3) a minimal system is either block $\mathcal{F}_{ip}$-sensitive for $\mu$ or an almost one-to-one extension of its $\infty$-step nilfactor.

    We also introduce the notion of topological $l$-sensitivity, and construct a minimal system which is $l$-sensitive but not $(l+1)$-sensitive for $l\in\mathbb{N}$.

    Mathematics Subject Classification: Primary: 37B05; Secondary: 37A05, 54H20.


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  • Figure 1.  $B^{(1)}, B^{(2)}$

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