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Let (X,f) be a dynamical system. In general the set of all ω-limit sets of f is not closed in the hyperspace of closed subsets of X. In this paper we study the case when X is a graph, and show that the family of ω-limit sets of a graph map is closed with respect to the Hausdorff metric.
We study the relation between transitivity and strong transitivity, introduced by W. Parry, for graph self-maps. We establish that if a graph self-map f is transitive and the set of fixed points of is finite for each k ≥ 1, then f is strongly transitive. As a corollary, if a piecewise monotone graph self-map is transitive, then it is strongly transitive.
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