Invariants of Twist - Wise Flow Equivalencemichael

Twist-wise ow equivalence is a natural generalization of ow equivalence that takes account of twisting in the local stable manifold of the orbits of a ow. Here we announce the discovery of two new invariants in this category. 1. Flow equivalence Square nonnegative integer matrices are used to describe maps on Cantor sets known as subshifts of nite type. Two such matrices are ow equivalent if their induced subshifts of nite type give rise to topologically equivalent suspension ows. Here topologically equivalent just means there is a homeomorphism, taking orbits to orbits, while preserving the ow direction. A matrix A is irreducible if for each (i; j) there is a power n such that the (i; j) entry of A n is nonzero. In terms of the corresponding subshift and suspension, irreducibility is equivalent to the existence of a dense orbit. Irreducible permutation matrices give rise to ows with a single closed orbit and are thus said to form the trivial ow equivalence class. For nontrivial irreducible incidence matrices John Franks has shown that ow equivalence is completely determined by two invariants, the Parry-Sullivan number and Bowen-Franks group. Let A be an n n incidence matrix. Then PS(A) = det(I ? A) and BF(A) = Z n (I ? A)Z n are the Parry-Sullivan number and the Bowen-Franks group respectively. See 8], 1], and 2] or the recent text 6]. Huang has settled the diicult classiication problem arising when the assumption of irreducibility is dropped, 3], 4], 5]. 2. Twist-wise flow equivalence Represent Z 2 by f1; tg, under multiplication with t 2 = 1. Let A(t) be an n n matrix with entries of the form a+bt, with a and b nonnegative integers. That is A is a matrix over the semigroup ring Z + Z 2. Call such a matrix a twist matrix. One interpretation of twist matrices is as follows. Suppose the suspension ow for A(1) is realized as a 1-dimensional basic set, B, of saddle type, of a ow on a 3-manifold. For each orbit in B there is a 2-dimensional local stable manifold, a ribbon, if you like. Call the union of such ribbons the ribbon set, and denote it by R. Each ribbon