Chaos of continuum-wise expansive homeomorphisms and dynamical properties of sensitive maps of graphs

Abstract

In this paper, we study several properties of chaos of maps of compacta. We show that if a homeomorphism / : X -» X of a compactum X with dimX > 0 is continuum-wis e expansive, then there is an /-invariant closed subset Y of X with dim Y > 0 such that / is (two-sided strongly) chaotic on Y in the sense of Ruelle-Takens . Also, we investigate dynamical properties of maps of graphs which are sensitive. In particular, we prove the decomposition theorem of sensitive maps of graphs as follows: If / : G ->• G is map of a graph G which is sensitive, then there exist finite subgraphs G% (1 1 such that Hi is /^-invariant , fn^\fk(Hi) : /*(#;) -> fk(Hi) (0 0} . As a corollary, we show that in case of maps of graphs, chaos in the sense of Ruelle-Takens is equal to (two-sided strongly) chaos in the sense of Devaney, and sensitive maps of graphs induce two-sided chaos in the sense of Li-Yorke.