Stronger Forms of Transitivity and Sensitivity for Nonautonomous Discrete Dynamical Systems and Furstenberg Families

Abstract

Let (Y, d) be a nontrivial metric space and (Y, g1,∞) be a nonautonomous discrete dynamical system given by sequences $(g_{l})_{l = 1}^{\infty }$ of continuous maps gl : Y → Y and let $\mathcal {F}$, $\mathcal {F}_{1}$ and $\mathcal {F}_{2}$ be given shift-invariant Furstenberg families. In this paper, we study stronger forms of transitivity and sensitivity for nonautonomous discrete dynamical systems by using Furstenberg family. In particular, we discuss the $\mathcal {F}$-transitivity, $\mathcal {F}$-mixing, $\mathcal {F}$-sensitivity, $\mathcal {F}$-collective sensitivity, $\mathcal {F}$-synchronous sensitivity, $(\mathcal {F}_{1},\mathcal {F}_{2})$-sensitivity and $\mathcal {F}$-multi-sensitivity for the system (Y, g1,∞) and show that under the conditions that gj is semi-open and satisfies gj ∘ g = g ∘ gj for each j ∈ {1, 2, ⋯ } and that $$\sum\limits_{j = 1}^{\infty}D(g_{j},g) $$exists (i.e., $\sum \limits _{j = 1}^{\infty }D(g_{j},g)<+\infty $), the following hold: The above results extend the existing ones.