Furstenberg families and transitivity in non-autonomous systems

We obtain necessary and sufficient conditions for a non-autonomous system to be [Formula: see text]-transitive and [Formula: see text]-mixing, where [Formula: see text] is a Furstenberg family. We also obtain some characterizations for topologically ergodic non-autonomous systems. We provide examples/counter examples related to our results.

Point transitivity, -transitivity and multi-minimality

Let $(X,f)$ be a topological dynamical system and ${\mathcal{F}}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an ${\mathcal{F}}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}:f^{n}(x)\in U\}$, is in ${\mathcal{F}}$; the system $(X,f)$ is called ${\mathcal{F}}$-point transitive if there exists some ${\mathcal{F}}$-transitive point. In this paper, we first discuss the connection between ${\mathcal{F}}$-point transitivity and ${\mathcal{F}}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by ${\mathcal{F}}$-point transitivity, completing results in [Transitive points via Furstenberg family. Topology Appl. 158 (2011), 2221–2231]. We also show that multi-transitivity, ${\rm\Delta}$-transitivity and multi-minimality can be characterized by ${\mathcal{F}}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates. Ergod. Th. & Dynam. Sys. 32 (2012), 1661–1672].

Furstenberg Families and Sensitivity

We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system(X,f)isℱ-sensitive if there exists a positiveεsuch that for everyx∈Xand every open neighborhoodUofxthere existsy∈Usuch that the pair(x,y)is notℱ-ε-asymptotic; that is, the time set{n:d(fn(x),fn(y))>ε}belongs toℱ, whereℱis a Furstenberg family. A dynamical system(X,f)is (ℱ1,ℱ2)-sensitive if there is a positiveεsuch that everyx∈Xis a limit of pointsy∈Xsuch that the pair(x,y)isℱ1-proximal but notℱ2-ε-asymptotic; that is, the time set{n:d(fn(x),fn(y))<δ}belongs toℱ1for any positiveδbut the time set{n:d(fn(x),fn(y))>ε}belongs toℱ2, whereℱ1andℱ2are Furstenberg families.