Study of chaotic maps and complicated invariant sets in topological dynamics by using continuum theory

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We investigated several dynamical properties of (continuum-wise) expansive homeomorphisms, invariant sets and minimal sets. R.Man'e proved that minimal sets of expansive homeomorphisms are 0-dimensional. More generally, we showed that minimal sets of continuum-wise expansive homeomorphisms are 0-dimensional. we proved that if f is a continuum-wise expansive homeomorphism of a compactum X with dim X=1, then there is a Cantor set Z in X such that for some natural number N,f^{N}(Z)=Z and f^{N}|Z is semi-conjugate to the shift homeomorphism of the symbolic dynamics. By using this result, we showed that there are many infinite minimal sets in X. Also, we investigated the topological entropy of maps on some fractal sets. Related to this study of dynamical systems, we also proved the following theorems in geometric topology : The sets of Bing maps and Lelek maps are G-delta dense in the mapping space of all maps from compacta to polyhedra. These results will be applied to the research of the theory ofdynamical systems.