ABSTRACT. Let (Xt,ot) be a shift of finite type, and G = aut(or) denote the group of homeomorphisms of Xt commuting with ctWe investigate the algebraic properties of the countable group G and the dynamics of its action on Xt and associated spaces. Using "marker" constructions, we show G contains many groups, such as the free group on two generators. However, G is residually finite, so does not contain divisible groups or the infinite symmetric group. The doubly exponential growth rate of the number of automorphisms depending on n coordinates leads to a new and nontrivial topological invariant of ot whose exact value is not known. We prove that, modulo a few points of low period, G acts transitively on the set of points with least or-period n. Using p-adic analysis, we generalize to most finite type shifts a result of Boyle and Krieger that the gyration function of a full shift has infinite order. The action of G on the dimension group of o~t is investigated. We show there are no proper infinite compact G-invariant sets. We give a complete characterization of the G-orbit closure of a continuous probability measure, and deduce that the only continuous G-invariant measure is that of maximal entropy. Examples, questions, and problems complement our analysis, and we conclude with a brief survey of some remaining open problems.

All common probability preserving transformations can be represented as elements of MPT, the group of measure preserving transformations of the unit interval with Lebesgue measure. This group has a natural Polish topology and the induced topology on the set of ergodic transformations is also Polish. Our main result is that the set of ergodic elements T in MPT that are isomorphic to their inverse is a complete analytic set. This has as a consequence the fact that the isomorphism relation is also a complete analytic set and in particular is not Borel. This is in stark contrast to the situation of unitary operators where the spectral theorem can be used to show that conjugacy relation in the unitary group is Borel. This result explains, perhaps, why the problem of determining whether ergodic transformations are isomorphic or not has proven to be so intractable. The construction that we use is general enough to show that the set of ergodic T 's with nontrivial centralizer is also complete analytic.On the positive side we show that the isomorphism relation is Borel when restricted to the rank one transformations, which form a generic subset of MPT. It remains an open problem to find a good explicit method of checking when two rank one transformations are isomorphic.In Memoriam: Prior to the final proofs of our paper our dear friend, Dan Rudolph, died just short of his 60th birthday, after a long and valiant struggle with ALS. He took an active part in this work to the very end despite his illness. This is not the place to describe in detail his lasting contributions to the modern theory of measurable dynamics; suffice it to say that his loss will be deeply felt not only by his close collaborators but by all who have an interest in this field. We want to dedicate our part of this paper to his memory.

We call an ergodic measure-preserving action of a locally compact group G on a probability space simple if every ergodic joining of it to itself is either product measure or is supported on a graph, and a similar condition holds for multiple self-joinings This generalizes Rudolph's notion of minimal self-joinings and Veech's property S Main results The joinings of a simple action with an arbitrary ergodic action can be explicitly descnbed A weakly mixing group extension of an action with minimal self-joinings is simple The action of a closed, normal, co-compact subgroup in a weakly-mixing simple action is again simple Some corollaries Two simple actions with no common factors are disjoint The time-one map of a weakly mixing flow with minimal self-joinings is prime Distinct positive times in a Z-action with minimal self-joinings are disjoint 0 Introduction and definitions The notion of minimal self-joinings for Z-actions was introduced in [Ru2] as a source of counter-examples In this paper we generalize this notion to what we call simple group actions and develop some general theory for these actions This allows us to broaden the repertoire of actions displaying this sort of behaviour We deal with actions of fairly general groups because it is convenient for our purposes and not much more difficult, but the main interest lies in Z and R-actions Most of our results are new even within the setting of Z-actionsWe consider a standard Borel space (X, B), that is there exists a complete separable metric on X such that B = B(X) is the cr-algebra of Borel sets generated by the corresponding topology on X (By the remarks on p 138 of [Ma2] one can assume that the metric on X is actually compact) Suppose that X is equipped with a Borel probability measure /J, and that G is a locally compact group By a (left) action of G on X we mean a Borel map GxX->X denoted (g, x)>-*gx such that (hg)x = h(gx)Vh,g€G,xeX, and ex = x Vx e X,

No abstract

Let p and q be relatively prime natural numbers. Define T o and S o to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1).Let (x. be a borel measure invariant for both T o and S o and ergodic for the semigroup they generate. We show that if /* is not Lebesgue measure, then with respect to fx. both T o and S o have entropy zero. Equivalently, both T o and S o are H-almost surely invertible.