Chaos For Topological Dynamics And Szemeredi-Type Theorems

It is well known that chaos is one of research hotspots in topological dynamics,Devaney and Li-Yorke chaos are most two popular definitions.Since Furstenberg gave the dynamical proof of the Szemeredi theorem in 1977,the study of combinatorial problems by ergodic theory has also become one of the key points.In this thesis,we mainly study the Devaney and Li-Yorke chaos for topological dynamics which under group or semigroup actions.Moreover,by using ergodic-theoretic techniques we prove some Szemeredi-type theorems.This paper is divided into four chapters.In Chapter 1,main results and preliminaries are stated.In Chapter 2,we introduce the concepts of Devaney and Li-Yorke chaos in topologi-cal dynamics under group or semigroup actions,give the definitions of muti-dimensional chaos and multi-dimensional Li-Yorke chaos,and derive some statements about De-vaney chaos implies Li-Yoreke chaos.Particularly,let R+(?)?X be a C0-semiflow on a Polish space,then we show:?If R+(?)?X is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior,then it is multi-dimensional Li-Yorke chaotic;that is,there is an uncountable set(?)C X such that for any k ?2 and any distinct points x1,...,xk?(?),one can find two time sequences sn??,tn??with sn(x1,...,xk)?(x1,...,xk)?Xk and tn(x1,...,xk)?(p,...,p)??xk.Consequently,Devaney chaos(?)Multi-dimensional Li-Yorke chaos.We in addition construct a completely Li-Yorke chaotic minimal SL(2,R)-acting flow on the compact metric space R? {?}.Our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup/group T.In Chapter 3,we mainly study the chaos of weakly-mixing flow or semiflow.Let X be a non-singleton Polish space,then we prove:?Any weakly-mixing C0-semiflow R+(?)?X is densely multi-dimensional Li-Yorke chaotic.? Any minimal weakly-mixing topological flow T(?)?X with T abelian is densely multi-dimensional Li-Yorke chaotic.?Any weakly-mixing topological flow T(?)?X is densely Li-Yorke chaotic.In Chapter 4,By using ergodic theoretic techniques following Furstenberg.we prove that measurable subsets of a locally compact abelian group of positive upper density contain Szemeredi-wise configurations defined by an arbitrary compact subset of the group.We mainly prove the following:?Let(G,+)be a locally compact Hausdorff abelian topological group and F C G a compact subset.If a measurable set E(?)G G is of positive upper density corresponding to an F-F(?)lner sequence F=(Fa)n=1? in(G,+,|·|),then for any g1,...,gl<F>,BD*({d?Z| DF*({u E E:u + d{g1,...,gl}(?)E})>0})>0.Here BD*denotes the lower Banach density of sets in(Z,+,|·|z)and(F)stands for the subgroup of G generated by F.A corollary of the above theorem is the following,which is is a generalization of the classical Szemeredi theorem for G = Z due to E.Szemeredi[50],for G = Zm due to Furstenberg and Katznelson[25],and for G = Rm with the Euclidean metric topology due to H.Furstenberg[23,Theorem 7.17].?Let(G,+)be a second countable locally compact Hausdorff abelian topological group.If a measurable set E C G is of positive upper Banach density,i.e.,BD*(E)>0,then for any g1,...,g,? G BD*({d ? Z | BD*({u?E:u + d{g1,...,gl}(?)E})>0})>0.