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Furstenberg families and sensitivity. (English) Zbl 1187.37016
Summary: We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system $(X,f)$ is ${\cal F}$-sensitive if there exists a positive $\varepsilon$ such that for every $x\in X$ and every open neighborhood $U$ of $x$ there exists $y\in U$ such that the pair $(x,y)$ is not ${\cal F}$-$\varepsilon$-asymptotic; that is, the time set $\{n:d(f^n(x),f^n(y))>\varepsilon\}$ belongs to ${\cal F}$, where ${\cal F}$ is a Furstenberg family. A dynamical system $(X,f)$ is $({\cal F}_1,{\cal F}_2)$-sensitive if there is a positive $\varepsilon$ such that every $x\in X$ is a limit of points $y\in X$ such that the pair $(x,y)$ is ${\cal F}_1$-proximal but not ${\cal F}_2$-$\varepsilon$-asymptotic; that is, the time set $\{n:d(f^n(x),f^n(y))<\delta\}$ belongs to ${\cal F}_1$ for any positive $\delta$ but the time set $\{n:d(f^n(x),f^n(y))>\varepsilon\}$ belongs to ${\cal F}_2$, where ${\cal F}_1$ and ${\cal F}_2$ are Furstenberg families.

37B05Transformations and group actions with special properties
37B20Notions of recurrence
54H20Topological dynamics
58K15Topological properties of mappings
37B10Symbolic dynamics
Full Text: DOI EuDML
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