Broken family sensitivity in transitive systems

Abstract

Let $(X,T)$ be a topological dynamical system, $n\ge 2$ and $\mathcal{F}$ be a Furstenberg family of subsets of ${\mathbb{Z}}_{+}$. $(X,T)$ is called broken $\mathcal{F}$-$n$-sensitive if there exist $\delta >0$ and $F\in \mathcal{F}$ such that for every opene (non-empty open) subset $U$ of $X$ and every $l\in \mathbb{N}$, there exist $x_1^l,x_2^l,\dots ,x_n^l\in U$ and $m_l\in {\mathbb{Z}}_{+}$ satisfying $d(T^k{x}_i^l,T^k{x}_j^l)>\delta ,\forall 1\le i<j\le n,k\in m_l+F\cap [1,l]$. We investigate broken $\mathcal{F}$-$n$-sensitivity for the family of all piecewise syndetic subsets (${\mathcal{F}}_{ps}$), the family of all positive upper Banach density subsets (${\mathcal{F}}_{pubd}$) and the family of all infinite subsets (${\mathcal{F}}_{inf}$). We show that a transitive system $(X,T)$ is broken $\mathcal{F}$-$n$-sensitive for $\mathcal{F}={\mathcal{F}}_{ps}\text{or}{\mathcal{F}}_{pubd}$ if and only if there exists an essential $n$-sensitive tuple which is an $\mathcal{F}$-recurrent point of $(X^n,T^{(n)})$; is broken ${\mathcal{F}}_{inf}$-$n$-sensitive if and only if there exists an essential $n$-sensitive tuple $(x_1,x_2,\dots ,x_n)$ such that ${\mathrm{limsup}}_{k\to \infty }{\min }_{1\le i<j\le n}d(T^k{x}_i,T^k{x}_j)>0$. We also obtain specific properties for them by analyzing the factor maps to their maximal equicontinuous factors. Furthermore, we show examples to distinguish different kinds of broken family sensitivity.

arXiv comment18 pages. J. Math. Anal. Appl. (2022), online

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