Complexities And Embeddings In Dynamical Systems

This thesis studies various complexities in topological dynamical systems on compact metric spaces.For zero entropy systems,we investigate topological complexity,se-quence entropy and entropy dimension.For positive entropy systems,we study the phenomenon of chaos inside them.For infinite entropy systems,we consider the em-bedding problem related to mean dimension.It consists of five parts.In the first part,we review some elementary notions and results on ergodic theory and dynamical systems,and collect all the tools which will be used in the whole thesis.The second part is devoted to the study of dynamical properties of group extensions over irrational rotations on the torus.We compute the topological complexity of these systems,prove the minimality by a new method,and give an equivalent characterization of these systems to be systems of order 2.Finally,we construct a minimal distal system which has linear topological complexity but is not a 2-step nilsystem,and thus give a negative answer to an open question raised by Host,Kra and Maass.The third part focuses on systems of zero sequence entropy.We investigate the relation of sequence entropy between a system and its induced system,and further study their relation of entropy dimension.More precisely,we prove that for a given sequence,a topological dynamical system has zero sequence entropy along this sequence if and only if so does its induced system.As an application,we get that the upper entropy dimension of a system is equal to that of its induced system.We also obtain the version of ergodic measure-preserving systems related to the sequence entropy and the upper entropy dimension.In the fourth part,we identify a family of sequences for which positive entropy implies multivariant mean Li-Yorke chaos for them.Examples are given for the classic sequences of primes and generalized polynomials.In the fifth part,we research the embedding problem related to the Lindenstrauss-Tsukamoto conjecture.We prove a topological version of Takens' theorem.As a corol-lary,we verify the Lindenstrauss-Tsukamoto conjecture for finite-dimensional topolog-ical dynamical systems.We also establish a new condition that implies embeddability of systems admitting factors of bounded Rokhlin dimension.