RESIDUAL BEHAVIOR OF INDUCED MAPS

Consider (X, F, mu, T) a Lebesgue probability space and measure preser ving invertible map. We call this a dynamical system. For a subset A i s an element of F, by T-A: A --> A we mean the induced map, T-A(x) = T -rA(x) where r(A)(x) = min{i > 0: T-i(x) is an element of A}. Such ind uced maps can be topologized by the natural metric D(A, A') = mu(A Del ta A') on F mod sets of measure zero. We discuss here ergodic properti es of T-A which are residual in this metric. The first theorem is due to Conze. THEOREM 1 (Conze): For T ergodic, T-A is weakly mixing for a residual set of A. THEOREM 2: For T ergodic, 0-entropy and loosely Be rnoulli, T-A is rank-1 and rigid for a residual set of A. THEOREM 3: F or T ergodic, positive entropy and loosely Bernoulli, T-A is Bernoulli for a residual set of A. THEOREM 4: For T ergodic of positive entropy , T-A is a K-automorphism for a residual set of A. A strengthening of Theorem 1 asserts that A can be chosen to lie inside a given factor al gebra of T. We also discuss even Kakutani equivalence analogues of The orems 1-4.