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Abstract. We prove that any topological conjugacy between subshifts is decomposed into the product of'bipartite codes', and obtain a natural generalization of Williams' theorem to sofic systems: two sofic systems are topologically conjugate iff the 'representation matrices' of the right [left] Krieger covers for them are 'strong shift equivalent' within right [left] Krieger covers; a similar result with respect to Fischer covers holds for transitive sofic systems. IntroductionWe call a 1-block factor map of a topological Markov chain onto a sofic system a cover for the sofic system or a sofic cover (cf. , a direct proof for Krieger's result with respect to Fischer covers has been given. In this paper, we are concerned with the above canonical covers and elucidate further topological conjugacy for sofic systems.First we introduce 'bipartite codes' and prove that any topological conjugacy between subshifts is decomposed into the product of bipartite codes. Using this result and using 'bipartite lambda graphs', we can prove results which include the above Krieger's results, and at the same time give a natural generalization of Williams' theorem [13] to sofic systems: we introduce 'representation matrices' and 'strong shift equivalence' for them and prqye that two sofic systems are topologically conjugate iff the representation matrices of the right [left] Krieger covers for them are strong shift equivalent within right [left] Krieger covers; a similar result is also proved for transitive sofic systems with respect to Fischer covers. In § 5, we give a practical method based on automata theory to construct the Krieger covers for a

Two results on textile systems are obtained. Using these we prove that for any automorphism φ of any topologically-transitive subshift of finite type, if φ is expansive and φ or φ−1 has memory zero or anticipation zero, then φ is topologically conjugate to a subshift of finite type. Moreover, this is generalized to a result on chain recurrent onto endomorphisms of topologically-transitive subshifts of finite type. Using textile systems and textile subsystems, we develop a structure theory concerning expansiveness with the pseudo orbit tracing property on directionally essentially weakly one-sided resolving automorphisms and endomorphisms of subshifts.

The global maps of homomorphisms of directed graphs are very closely related to homomorphisms of a class of symbolic dynamical systems called subshifts of finite type. In this paper, we introduce the concepts of ‘induced regular homomorphism’ and ‘induced backward regular homomorphism’ which are associated with every homomorphism between strongly connected graphs whose global map is finite-to-one and onto, and using them we study the structure of constant-to-one and onto global maps of homorphisms between strongly connected graphs and that of constant-to-one and onto homomorphisms of irreducible subshifts of finite type. We determine constructively, up to topological conjugacy, the subshifts of finite type which are constant-to-one extensions of a given irreducible subshift of finite type. We give an invariant for constant-to-one and onto homomorphisms of irreducible subshifts of finite type.

Abstract. In this paper, we investigate some combinatorial aspects of C-surjective local maps, i.e., local maps inducing surjective global maps, CF-surjective local maps, i.e., local maps inducing surjective restrictions of global maps on the set CF of finite configurations, and C-injective local maps, i.e., local maps inducing injective local maps, of one-dimensional tessellation automata.We introduce a pair of right and left bundle-graphs and a pair of right and left X-bundle-graphs for every C-surjective local map. We give characterizations for CF-surjectivity, C-injectivity and some other properties of C-surjective local maps in relation to these bundle-graphs. We also establish some properties of the inverse of a C-injective local map.