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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 $${\mathcal F}$$-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 $${\mathcal F}$$-$$\varepsilon$$-asymptotic; that is, the time set $$\{n:d(f^n(x),f^n(y))>\varepsilon\}$$ belongs to $${\mathcal F}$$, where $${\mathcal F}$$ is a Furstenberg family. A dynamical system $$(X,f)$$ is $$({\mathcal F}_1,{\mathcal F}_2)$$-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 $${\mathcal F}_1$$-proximal but not $${\mathcal F}_2$$-$$\varepsilon$$-asymptotic; that is, the time set $$\{n:d(f^n(x),f^n(y))<\delta\}$$ belongs to $${\mathcal F}_1$$ for any positive $$\delta$$ but the time set $$\{n:d(f^n(x),f^n(y))>\varepsilon\}$$ belongs to $${\mathcal F}_2$$, where $${\mathcal F}_1$$ and $${\mathcal F}_2$$ are Furstenberg families.

##### MSC:
 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 37B20 Notions of recurrence and recurrent behavior in dynamical systems 54H20 Topological dynamics (MSC2010) 58K15 Topological properties of mappings on manifolds 37B10 Symbolic dynamics
##### Keywords:
Furstenberg families; sensitivity; symbolic dynamics
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##### References:
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