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Restricted orbit changes of ergodic ℤd-actions to achieve mixing and completely positive entropy

Adam Fieldsteel, N. A. Friedman

1986
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Ergodic Theory and Dynamical Systems
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We show that for every ergodic Z d -action T, there is a mixing Z d -action S which is orbit equivalent to T via an orbit equivalence that is a weak a-equivalence for all a > 1 and a strong fc-equivalence for all b e (0,1). If T has positive entropy, then S can be taken to have completely positive entropy. If the dimension d is greater than one, the orbit equivalence may be taken to be bounded and a strong b-equivalence for all b > 0. Let T and 7" be ergodic, measure-preserving Z d -actions on
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... ebesgue probability spaces (ft, 38, /JL) and (ft', 38', /A')-T and T" are said to be orbit equivalent if there exists a non-singular, bimeasurable map : ft -> ft', such that, for a.e. (o e ft, (f> maps the T-orbit O T {u>) of w bijectively onto O T -((tii)). Equivalently, and more conveniently for our purposes, T and T are orbit equivalent if there is a Z d -action S on ft, isomorphic to I", having the same orbits as T. That is, for a.e. a>, O T (a)) = O S (CD). We refer to such a pair (T, S) as an orbit equivalence between T and S. Given an orbit equivalence between T and S, there is a measurable function a:ftxZ d H>Z d given by, for all veZ d and for a.e. w, T v (co) = S" (M>I)) («), and satisfying (i) a(w, •);Z d -*Z d is a bijection for a.e. w and (ii) a(w, u+w) = a(&), u) + a(T"w, w), for a.e. w and all v, weZ d . We refer to a as the cocycle of the orbit equivalence (T, S). Conversely, given such a cocycle a for T (that is, a function a satisfying (i) and (ii)) we can define a function a~':ftxZ d -*Z d by a (a, a"'(w, u)) = i > and a Z d -action S by 5 u (w)= r a~W ) ( « ) so that (T, S) is an orbit equivalence with cocycle a. (Note that a" 1 is the cocycle for (S, T).) Thus we can (informally) regard orbit equivalences and the associated cocycles as the same objects. We will indicate the above relationship between T and S by S = T" '. We remark that a theorem of Dye [2] asserts that every two (ergodic, Lebesgue probability measure-preserving) Z d -actions are orbit equivalent. On the other extreme, we can describe isomorphism of Z d -actions by saying that T is isomorphic to S if and only if there is an orbit equivalence between T and S with cocycle a satisfying for all v, a.e. l) that is carried out in [1] . There one finds a family of relations, -* M , parametrized by M 6 GL(n, R). The relation corresponding to the parameter ideGL(n, R) can be defined using the statement of the above theorem as a model.

doi:10.1017/s0143385700003667
fatcat:fhj474iwlvh7zcyx6iee5u4zdm