Dynamic Properties Of G-Maps

Topological dynamical system is a very important branch of nonlinear science, and widely applied to physics, biology, economics and other subjects. In this thesis, we mainly study the dynamic properties of G-maps and their induced maps. The main contents of this thesis are as follows.In the introduction, we introduce the origin, developments and research status of dynamic systems.In Chapter2, we mainly recall some basic concepts of metric spaces and introduce some basic concepts of G-maps and some important lemmas.In Chapter3, we study the asymptotic properties of the G-maps. We study the properties of the G-ω-limit sets, the G-almost periodic point sets, the G-recurrent sets, the G-chain recurrent sets and the G-non-wandering sets of G-maps respectively. Moreover, we prove that PG(f)∈APG(f)∈RG(f)∈ΩG(f)∈CRG(f).In Chapter4, we study the density of G-maps in the phase space. Firstly, we study some equivalent conditions for which the maps are G-transitive. Secondly, we study some properties and two equivalent conditions of G-weak- ly mixing.In Chapter5, we discuss the dynamic properties of set-valued maps under the group action. Firstly, the following three statements are equivalent:(1)f is G-weakly mixing;(2)f is G-weakly mixing;(3)f is G-transitive. Secondly, we discuss the G-sensitive and G-chaos. Under the We-topolo-gy, then f being G-chaotic in the sense of Devaney implies f being G-ch-aotic in the sense of Devaney.In Chapter6, we give an example to show that even if f is G-transitive, f is not always transitive; and even if f is G-mixing,f is not always G-w-eakly mixing.