An Algebraic Obstruction to Isomorphism of Markov Shifts with Group Alphabets

Given a compact group G, a standard construction of a Z2 Markov shift ΣG with alphabet G is described. The cardinality of G (if G is finite) or the topological dimension of G (if G is a torus) is shown to be an invariant of measurable isomorphism for ΣG. We show that if G is sufficiently non–abelian (for instance A5, PSL2(F7) or a Suzuki simple group) and H is any abelian group with |H| = |G|, then ΣG and ΣH are not isomorphic. Thus the cardinality of G is seen to be necessary but not sufficient to determine the measurable structure of ΣG. 1991 Mathematics Subject Classification. 22D40, 28D20. Supported in part by N.S.F. grant No. DMS–91–03056 at the Ohio State University Typeset by AMS-TEX 1 §1: Introduction Let G be a compact group with normalised Haar measure μG. Let XG = {x ∈ GZ 2 | x(i,j) = x(i,j−1) · x(i+1,j−1) for all i, j ∈ Z}. (1) An element x of XG is determined by specifying the coordinates x(n,0) and x(0,m) for n ∈ Z and m ∈ N>0, so Haar measure μG ⊗ μG on the compact group GZ × GN determines a probability measure λG on XG, defined on the class of subsets BG of XG determined by measurable subsets of GZ ×GN. If G is abelian, then XG is a subgroup of GZ 2 and in this case λG is Haar measure on the group XG. The probability space (XG,BG, λG) supports a natural representation α of Z by λG– preserving transformations of XG via the shift action (α (n,m)x)(i,j) = x(n+i,m+j). (2) The dynamical system ΣG = (α, XG) is, to within a trivial change, a generalisation of the “3–dot” or “1+x+y” system considered with various possible choices of G in [5], [6], [7], [9], [13] and [14]. We summarize the most important properties here. If G = T, the circle group, then XG is a connected compact group, and the dynamical system ΣG is an action by automorphisms with the Descending Chain Condition [5, Defn. 3.1]. It follows that the periodic points in ΣG are dense [5, Thm. 7.2]. The action αG has completely positive entropy [9, Thm. 6.5] and is therefore mixing of all orders [3, Thm. 2]. In fact ΣG is measurably isomorphic to a Z Bernoulli shift [13, Thm. 2.4]. If G = {0, 1}, the group with two elements, then ΣG again has the Descending Chain Condition (equivalently, αG acts expansively – see [5, Thm. 5.2]) and dense periodic points ([5, Thm. 11.6]). This example (and other related actions) was shown by Ledrappier in 1978 to be a zero–entropy Markov shift which is mixing but not 2–fold mixing [7]. The periodic points are not uniformly distributed: along certain sequences of periods, the periodic points are uniformly distributed with respect to Haar measure, while along other sequences of periods there is only one periodic point for each period [14, Ex.3.3]. The shape {(0, 0), (1, 0), (0, 1)} is a minimal non–mixing shape [11, Ex.7.13] in the sense of [6] (see also §3). The failure of 2–fold mixing and the bad distribution of periodic points are manifestations of the same phenomenon: ΣG does not have any specification properties. If G is finite, then the measure λG may be described very easily on cylinder sets defined on contiguous sets. If E ⊂ Z is a finite contiguous set of positions, define an E–cylinder set by AE = {x ∈ XG | x(n,m) = a(n,m) for (n,m) ∈ E} where {a(n,m) | (n,m) ∈ E} is an allowed word satisfying (1). Let n(E) denote the number of positions in E at which we may choose the value of x independently: that is, the (n(E)+1) position is the first one to be determined by the preceding n(E) positions. Then λG(AE) = 1/|G|. For instance, if E = {(0, 0), (0, 1), (1, 0), (1, 1)} then n(E) = 3 because we can