Application of Infinite Labeled Graphs to Symbolic Dynamical Systems



We apply a theory of infinite labeled graphs to studying presentations and classifications of symbolic dynamical systems, by introducing a class of infinite labeled graphs, called λ-graph systems. Its matrix presentation is called a symbolic matrix system. The notions of a λ-graph system and symbolic matrix system are generalized notions of a finite labeled graph and symbolic matrix for sofic subshifts to general subshifts. Strong shift equivalence and shift equivalence between symbolic matrix systems are formulated so 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 K-groups, and the Bowen–Franks groups that are generalizations of the corresponding notions for nonnegative matrices. They yield topological conjugacy invariants of subshifts. The entropic quantities called λ-entropy and volume entropy for λ-graph systems are also studied related to the topological entropy of symbolic dynamics.


Subshifts Symbolic dynamics λ-Graph systems Strong shift equivalence Bowen–Franks group K-theory Topological entropy 


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The author would like to deeply thank Matthias Dehmer and Jun Ichi Fujii for their invitation to the author to write this chapter and for their helpful suggestions in the presentation of this paper.


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsJoetsu University of EducationJoetsuJapan

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