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Subjects
37-XX Dynamical systems and ergodic theory
37Exx Low-dimensional dynamical systems
37E25 Maps of trees and graphs

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Spaces of ω-limit sets of graph maps

Jie-Hua Mai, Song Shao (2007)

Fundamenta Mathematicae

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.

Strong Transitivity and Graph Maps

Katsuya Yokoi (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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 $f^{k}$ 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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