Topological conjugacy for sofic systems

Abstract

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

The Syntactic Graph of a Sofic Shift

“…Nasu's Classification Theorem holds for reducible sofic shifts by the use of right Krieger covers instead of right Fischer covers [19]. This enables the extension of our result to the case of reducible sofic shifts.…”

“…[19] we can assume, without loss of generality, that the symbolic adjacency matrices of the right Fischer covers of X and Y are elementary strong shift equivalent. We define the bipartite shift Z as above.…”

“…An effective procedure to construct the right Krieger cover of a sofic shift is described in [19]. First, one constructs the (unique) minimal deterministic automaton with one initial state recognizing the language of finite blocks of the shift.…”

“…The proof of the conjugacy invariance is based on Nasu's Classification Theorem for sofic shifts [19] that extends William's one for shifts of finite type. This theorem says that two irreducible sofic shifts X, Y are conjugate if and only if there is a sequence of symbolic adjacency matrices of right Fischer covers A = A 0 , A 1 , .…”

“…This means that, for each i, there are two symbolic matrices U i and V i such that, after recoding the alphabets of A i−1 and A i , we have A i−1 = U i V i and A i = V i U i . A bipartite shift is associated in a natural way to a pair of elementary strong shift equivalent and irreducible sofic shifts [19].…”