The product symbolic dynamical systems. (English) Zbl 1171.37304
Summary: A product symbolic dynamical system (PSDS) is more complex than the ordinary symbolic dynamical systems (OSDS), serves as a universal system for compact and totally disconnected dynamical systems, and provides a moderate framework for characterizing orbit structures of general dynamical systems, in particular for non-expansive dynamical systems. This paper first explores fundamental properties of PSDS: although it shares some properties of 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). Then, a necessary and sufficient condition for the existence of 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 provides useful a tool for identifying more complicated invariant sets of the given dynamical systems. Moreover, a general characterization of subshifts is also established for the PSDS, which reveals the structures of all compact and totally disconnected dynamical systems.
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