Almost topological classification of finite-to-one factor maps between shifts of finite type

Abstract

Abstract. We classify finite-to-one factor maps between shifts of finite type up to almost topological conjugacy. IntroductionIn [2], shifts of finite type (SFT) were classified up to almost topological conjugacy. The purpose of this paper is to classify finite-to-one continuous factor maps between shifts of finite type. We consider two equivalence relations. The first is very strong: two maps are equivalent if they have the same range shift and they differ by a special kind of almost continuous change of variables in the domain; for a fixed range shift and a fixed generic cardinality of the fibre of the factor maps (a natural invariant), we get infinitely many equivalence classes. The second (more natural) equivalence relation allows the same kind of change of variables in the range (as well as in the domain) and gives only finitely many equivalence classes (for a fixed entropy class and generic cardinality of the fibre). For the latter classification, we reduce the problem to a group action problem which was solved by us in [1]. Our results are completely analogous to and were inspired by those of D. Rudolph [10] in the measure-theoretic category (i.e. classification of finite-to-one factor maps between Bernoulli shifts). Our work is basically a more concrete version of Rudolph's work. In particular, let T be a Bernoulli shift whose measure-theoretic entropy (log (A)) is the same as the topological entropy of an aperiodic SFT; then for any finite-to-one factor map of T, we construct (theorem 4.3) an equivalent (in the measure-theoretic sense) factor map w:2 A ->S B , where 2 B is an arbitrary aperiodic SFT of entropy log (A) and S A is some aperiodic SFT of entropy log (A). (In general, 2 A cannot be chosen arbitrarily.) It would perhaps have been most natural for us to classify our factor maps with respect to continuous changes of variables (i.e. topological conjugacy) since, after all, our factor maps are continuous themselves. But the simplest case of this, when the factor maps involved are identity maps, reduces to the classical topological conjugacy problem for shifts of finite type-which is not yet solved satisfactorily.For background, we basically refer to [2]. We use the notation 2 A for an SFT defined by 0-1 transition matrix A, cr A for the shift, and L A for the alphabet (or

Decompositions of factor maps involving bi-closing maps

“…This result can be viewed as a generalization of [4, Theorem 9 and Corollary 10]. P. TRow Example 2 in [1] gives a finite-to-one factor map which is not almost topologically conjugate over its range to a bi-closing map. On the other hand, it follows from [7,Theorem 4.1] that any finite-to-one factor map is almost topologically conjugate over its range to a right closing map, and to a left closing map (see Theorem 1.2).…”

“…In [1], ADLER, KITCHENS and MARCUS classified finite-to-one factor maps between shifts of finite type, up to certain equivalence relations. They also gave an example of a factor map 0, of degree d, which is impossible to decompose into two maps, 0 = '7~b, such that "7 is d-to-1 everywhere ([1, Example 2, p. 492]).…”

“…If we require that deg("7) have a specified value greater than one, then such a decomposition may not exist, as Example 2 in [1] shows. In particular, if we require that deg("7) ----deg(0), then the existence of such a decomposition is equivalent to the statement that 0 is almost topologically conjugate over S to a bi-closing map.…”

“…The maps 01 and 02 are almost topologically conjugate over S if there exists an irreducible shift of finite type ~o and one-to-one almost everywhere factor maps ~i (i = 1,2) such that 01~bl = 02~b2 (see [1, p. 487] Proof Since 02 is finite-to-one, it is easy to see that ~b~ is finite-to-one. Since Ee, is irreducible, it follows from [6,Theorem 3.3] that ~bl I~E, is onto (see the proof of [1,Theorem 9]). To see that ~311~ e, is one-to-one almost everywhere, suppose that x E Ea~ is doubly transitive.…”

Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings

“…We will prove that (2) implies (1). As shown by Parry [27], the given conjugacy ' can be given as a string of state splittings from ðAÞ to some C followed by the reversal of a string of state splittings from ðBÞ to C. The SSEs over Z þ that give the splittings are easily adapted to SSEs over Z þ G which reect the corresponding lifting of edge labelings (we give an example following the proof).…”