Research of invariant sets of topological dynamics in continuum theory

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We investigated some properties of invariant sets and minimal sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms, which are important in Chaotic dynamical systems. In 1979, Mane proved that every minimal set of expansive homeomorphism is 0-dimensional, and the head investigator generalized this result concerning to continuum-wise expansive homeomorphism. In this rearch project, we proved that each continuum-wise expansive homeomorphism of 1-dimensional compactum admits many minimal sets which are Cantor sets. Also, every continuum-wise fully expansive homeomorphism homeomorphism admits many minimal sets which are Cantor sets. These rtesults imply that ( continuum-wise ) expansive homeomorphisms induce chaos of periodicity.