Measurable sensitivity via Furstenberg families

Tao Yu

doi:10.3934/dcds.2017194

Article Contents

2017, Volume 37, Issue 8: 4543-4563. Doi: 10.3934/dcds.2017194

This issue Previous Article Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity Next Article Ground state solutions for Hamiltonian elliptic system with inverse square potential

Measurable sensitivity via Furstenberg families

Tao Yu^,

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

^* Corresponding author: Tao Yu
^* Corresponding author: Tao Yu

Received: September 2016

Revised: March 2017

Published: May 2017

The author was supported by NNSF of China (11371339,11431012,11571335).

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Abstract

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 $l\in\mathbb{N}$.

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

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