Dynamics of homeomorphisms of regular curves

2018/06/30 by Naghmouchi, Issam · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.00222

Abstract

In this paper, we prove first that the space of minimal sets of any homeomorphisms f:X→ X of a regular curve X is closed in the hyperspace 2^X of closed subsets of X endowed with the Hausdorff metric, and the non-wandering set Ω(f) is equal to the set of recurrent points of f. Second, we study the continuity of the map ω_f:X→ 2^X;x↦ ω_f (x) , we show for instance the equivalence between the continuity of ω_f and the equality between the ω-limit set and the α-limit set of every point in X . Finally, we prove that there is only one (infinite) minimal set when there is no periodic point.

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