Kolyada S. Topological dynamics: minimality, entropy and chaos.

We investigate topologically transitive and minimal dynamical systems on Hausdorff compact spaces. It is shown that any minimal map is almost open (and if it is even open then it is a homeomorphism). The Li-Yorke definition of chaos proved its value for interval maps, we considered it in the setting of general topological dynamics. We solved a long-standing open question by proving that positive entropy implies Li-Yorke chaos. We introduced and studied a new concept of chaos -- Li-Yorke sensitivity. It is proved that for any weak mixing dynamical system, the proximal cell of any point is dense in the space. The basic properties of triangular continuous maps and topological entropy of nonautonomous dynamical systems are investigated. We gave two closely related axiomatic definitions of topological entropy and an axiomatic characterization of the topological chaos