Based on these results, several new criteria of chaos for maps are finally established via strict coupled-expansions for irreducible transitive matrices in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, respectively, where their conditions are weaker than those existing in the literature. Dynamical behaviors of subshifts for irreducible matrices are then studied and several equivalent statements to chaos are obtained especially, chaos in the sense of Li-Yorke is equivalent to chaos in the sense of Devaney for the subshift, and is also equivalent to that the domain of the subshift is infinite. A certain relationship between strictly coupled-expanding maps for a transitive matrix in disjoint bounded and closed subsets of a complete metric space and their topological conjugacy to the subshift for the matrix is also obtained. It is found that the subshift for a transitive matrix is strictly coupled-expanding for the matrix in certain disjoint compact subsets the topological conjugacy of a continuous map in its compact invariant set of a metric space to a subshift for a transitive matrix has a close relationship with that the map is strictly coupled-expanding for the matrix in some disjoint compact subsets. For any y 2 X nU there exist disjoint clopen sets A. A probability measure on (the Borel -algebra of) or a subshift is called invariant if for all measurable G and all. quantities related to subshifts that stay invariant under conjugacy. One approach to solve the problem is by the study of invariants, i.e. One important question is to determine whether two SFTs are conjugate. SFTs are one of the most fundamental objects in symbolic dynamics LM95. A subshift X is called minimal if for all. A multidimensional shift of nite type (SFT) is a set of colorings of Z2 given by local rules. The orbit of is the set and the orbit closure of x is the closure of in. dimensional systems onto given higher dimensional systems.) A necessary condition for T to admit a subshift cover is that T must have finite entropy. The concept of coupled-expanding map is extended to a more general one: coupled-expansion for a transitive matrix. A subshift can be also given as the orbit closure of a sequence. A compact invariant subset of is called a subshift. In particular, this means that the stabilized automorphism groups on 2 symbols and 6 symbols are not isomorphic the analog of this result for the (non-stabilized) automorphism groups on 2 and on 6 symbols remains open.
This paper studies relationships between coupled-expanding maps and one-sided symbolic dynamical systems.
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