Dynamical Property Of Product Space And The Inverse Limit Space Of A Topological Group Action

At present, it is very perfect to development of theory and result of dynamical system of one dimensional interval space and the inverse limit space But in pratical application, mathematical model of many subjects mainly belongs to iterative problems of self-map of high dimensional product space. At the same time, many scholars also meet the problem of dynamics in the inverse limit space of a topological group action. Hence it makes sense to study dynamical property of high dimensional product space and the inverse limit space of a topological group action. In this paper, we will further study these two aspects.In Chapter3, we mainly gave12equivalent conditions of product map with closed periodic set in the n-dimensional product space.In Chapter4, we will study Devaney G-chaos and G-korner property of shift map in the inverse limit space of a topological group action. We have the following result:(1) Let (X, d) be a compact metric G-space, f:Xâ†'X be a homeomorphism map. The system (Xf,G, d,σ) is the inverse limit space of the system(X,G,d,f). Then f is said be G-chaotic in the sense of Devaney if and only ifσis said be G-chaotic in the sense of Devaney.(2) Let(X,d)be a compact metric G-space,f:Xâ†'Xbe a surjection. The system(Xf, G, d,σ) is the inverse limit space of the system (X,G,d,f). Then f is said be G-korner if and only if a is said be G-korner.In Chapter5, we will consider G-shadowing property and G-strong shadowing property of shift map in the inverse limit space of a topological group action. We have the following result:(1) Let(X,d)be a compact metric G-space,f:Xâ†'X be a surjective map. The system (Xf,G, d,σ)is the inverse limit space of the system (X,G,d,f).Then f has the G-shadowing property if and only ifσ has the G-shadowing property.(2) Let(X,d)be a compact metric G-space,f:Xâ†'X be a homeomorphism with the lipschitz constant L. f-1has the lipschitz constant K. The system (Xf,G, d,σ)is the inverse limit space of the system(X,G,d,f). Then f has the G-strong shadowing property if and only ifσ has the G-strong shadowing property.