Product Symbolic Dynamical Systems

Dynamical system is an essential component of nonlinear science. It investigates the limit behavior of the natural phenomena's evolution as time goes to infinity. Due to the fundmental work of Poincare, Lyapunov, Birkhoff, and so on, dynamical system appears as an important branch of the modern mathematics. It's well-known that the ordinary symbolic dynamical systems (OSDS) plays an important role in the investigation of dynamical system, because it does not only provide an useful tool for studying general dynamical systems, but also OSDS themselves are important subjects of the dynamical investigation. However, why do we investigate the product symbolic dynamical system (PSDS)? Firstly, a PSDS, which inherits the typical dynamical properties of the OSDS, is more complex than the OSDS, so it can be used to characterizing more general dynamical systems. Secondly, PSDS serves as a universal system for compact and totally disconnected dynamical systems (since OSDS cann't), and provides a moderate framework for characterizing orbit structures of general dynamical systems, in particular non-expansive dynamical systems. In this thesis, the systematic definition of PSDS is introduced, some dynamical properties of the PSDS are studied, generalized invariant set of shift (with respect to the PSDS) is defined, a necessary and sufficient condition for the existence of generalized invariant sets of shift (with respect to the PSDS) is established for the general dynamical systems, and another necessary and sufficient condition that determines exactly which of the topological dynamical system can be embedded into the PSDS is established, also subshifts of the PSDS is defined. The contents of the thesis are outlined as below:In Chapter 1, we mainly introduce the history of the PSDS, and give a physical description of the PSDS. The purpose and main results are also summarized in this chapter.In Chapter 2, we firstly give the systematic definition of the PSDS, then investigate fundamental properties of the PSDS, from which we conclude that every OSDS can be topologically embedding into the PSDS. although it shares some properties of the OSDS, such as dense periodic points and topological mixing, the PSDS holds other remarkable properties, such as non-expansivity, uncountable periodic points and infinite entropy. (the OSDS is expansive with countable periodic points and finite entropy) This shows the PSDS is more complex than the OSDS, so it can contribute to get further insights in the dynamics of complex systems.In Chapter 3, the generalized invariant sets of shift (with respect to the PSDS) is defined, and a necessary and sufficient condition for the existence of generalized invariant sets of shift (with respect to the PSDS) is established for the general dynamical systems, which generalizes the corresponding result on the invariant sets of shift (with respect to an OSDS), and also provides a useful tool for identifying more complicated invariant sets of the given dynamical systems.In Chapter 4, we give the definition of subshift (with respect to the PSDS) and an exclusion system , then investigate the relationship between them, which inherit the relationship between subshift (with respect to the OSDS) and an exclusion system. The general characterization of subshifts developed in this section reveals the structures of all compact and totally disconnected systems.In Chapter 5, a necessary and sufficient condition that determines exactly which of the topological dynamical system can be embedded into the PSDS is established, and it provides a facilitate device for identify the totally disconnect property of a topological space. In particular, the sufficiency implies a recent result by Akashi: every compact and totally disconnected dynamical system, expansive or not, can be conjugately embedded into a PSDS.In Chapter 6, we summarize the results and innovation of this thesis, and present our view of perspectives for the future investigation of the PSDS.