Michael Sullivan (Southern Illinois University)

Flow equivalence of skew-products of irreducible shifts of finite type

Abstract. Let G be a finite group, and F be a function from the edge set of a given irreducible shift of finite type (SFT) X into G. This can be thought of as a skew-product system or a G-weighted SFT. A G-weighted SFT is determined by a square matrix over the semi-group ring Z₊G. The following are known (W. Parry) to be equivalent conditions on two such matrices A and B arising from functions F and H, respectively,
(1) A and B determine isomorphic skew product systems,
(2) A and B are strong shift equivalence (SSE) over Z₊G,
(3) there exists a topological conjugacy of their associated SFTs which sends F to a function cohomologous to H, and
(4) (if G is abelian) there exists a topological conjugacy of their associated SFTs which respects G-weights on periodic orbits.

A and B define flow equivalent skew products if their skew products can be made isomorphic after a time change. In the case G=Z/2Z, the equivalence relation is called twistwise flow equivalence, which has been applied to understand the twisting in the local stable manifolds of basic saddle sets in Smale flows. The twistwise flow equivalence classification has been open for five years.

Although complete written proofs have not yet been finished for checking, we think we have a proof of the following

Theorem. Let G be a finite group. Let A and B be square matrices over Z₊Gdefining skew products of irreducible SFTs with G. Assume these SFTs are not trivial (i.e. contain more than one orbit). Then the skew products are flow equivalent if and only if the matrices I-A and I-Bare SL(ZG) equivalent, i.e., there exist matrices U, V over ZG with determinant 1 such that U(I-A)V = I-B.

This theorem grows out of the "positive K theory" framework in symbolic dynamics which has been applied in several categories. In fact the theorem is stronger: Any given SL equivalence (U,V) from I-A to I-B can be decomposed into a string of "positive" equivalences which yield flow equivalences.

Complete invariants for the algebraic relation of SL(ZG) equivalence are not hard to compute for G = Z/2 but the algebra becomes sophisticated even for G = Z/n. Expert algebraists know a lot about this (maybe everything).

We work in the setting of infinite, finitely supported matrices A(and the infinite identity I).