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Some Variations of Transitivity for CR-dynamical systems

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arxiv 2305.13813 v1 pith:4FGDW4JJ submitted 2023-05-23 math.DS

classification math.DS
keywords closeddynamicsmapsrelationrelationstransitivityapproachbullet
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abstract

We consider the topological dynamics of closed relations(CR) by studying one of the oldest dynamical property - `transitivity'. We investigate the two kinds of (closed relation) CR-dynamical systems - $(X,G)$ where the relation $G \subseteq X \times X$ is closed and $(X,G, \bullet)$ giving the `suitable dynamics' for a suitable closed relation $G$, where $X$ is assumed to be a compact metric space without isolated points. $(X,G)$ gives a general approach to study initial value problems for a set of initial conditions, whereas $(X,G, \bullet)$ gives a general approach to study the dynamics of both continuous and quasi-continuous maps. We observe that the dynamics of closed relations is richer than the dynamics of maps and find that we have much more versions of transitivity for these closed relations than what is known for maps.

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Cited by 1 Pith paper

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  1. Shadowing in CR-Dynamical Systems

    math.DS 2025-05 accept novelty 5.0 of 10

    Defines (i,j)-shadowing for closed relations on compact spaces and gives characterizations for finite-branching, diagonal-containing, and isometry-related relations.

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