Sofic Subshifts Research Articles

Sofic subshifts and piecewise isometric systems

We study the natural symbolic dynamics associated with piecewise continuous, non-invertible, dynamical systems. Our study is centered primarily on the relationship between the point-set topological properties of the partition of the system and the symbolic coding. We prove that for a class of maps locally preserving distances with regular partition, the associated symbolic dynamics cannot embed subshifts of finite type of positive entropy. Hence, in particular, almost sofic subshifts obtained from the symbolic dynamics have zero entropy. However, there are examples in Euclidean spaces of systems with non-regular partitions for which the coding maps can be surjective, particularly embedding all subshifts. For all such examples, the associated group of isometries is a subgroup of $O(\mathbb{R}, N)$.

Presentations of subshifts and their topological conjugacy invariants

We introduce the notions of symbolic matrix system and \lambda -graph system that are presentations of subshifts. They are generalized notions of symbolic matrix and \lambda -graph for sofic subshifts to general subshifts. We then formulate strong shift equivalence and shift equivalence between symbolic matrix systems and show that two subshifts are topologically conjugate if and only if the associated canonical symbolic matrix systems are strong shift equivalent. We construct several kinds of shift equivalence invariants for symbolic matrix systems. They are the dimension groups, the Bowen-Franks groups and the nonzero spectrum that are generalizations of the corresponding notions for nonnegative matrices. The K-groups for symbolic matrix systems are introduced. They are also shift equivalence invariants and stronger than the Bowen-Franks groups but weaker than the dimension triples. These kinds of shift equivalence invariants naturally induce topological conjugacy invariants for subshifts.

Formal languages and global cellular automaton behavior

A connection is made between the description of one-dimensional cellular automata as dynamical systems acting on a compact metric space, and as computational systems acting on strings. Operators are defined which determine a bijective correspondence between subshifts of infinite configurations and languages of finite strings. Under this correspondence, families of languages give rise to families of subshifts. Regular languages correspond to sofic subshifts. Families of subshifts corresponding to context-free, context-sensitive, and recursively enumerable languages are introduced and their closure properties under forward and backward cellular automaton images studied.

An invariant for bounded-to-one factor maps between transitive sofic subshifts

AbstractWe define a ‘core-matrix’ of a transitive sofic subshift, which is unique up to similarity for each transitive sofic subshift. We prove that if there exists a bounded-to-one factor map from one transitive sofic subshift to another, the block of the Jordan form of a core-matrix of this second subshift with non-zero eigenvalues is a principal submatrix of the Jordan form of a core-matrix of the first. We also prove that the subshifts that are almost of finite type are ‘spectrally of finite type’.