ISRAEL JOURNAL OF MATHEMATICS 93 (1996),387-398 RESIDUAL BEHAVIOR OF INDUCED MAPS BY ANDRES DEL JUNCO Department of Mathematics, University of Toronto Toronto, Ontario, Canada M55 IA1 e-mail: deljunco@math.toronto.edu AND DANIEL J. RUDOLPH Department of Mathematics, University of Maryland College Park, MD 20742, USA e-mail: djr@grace.umd.edu ABSTRACT Consider (X, 5 r, ~, T) a Lebesgue probability space and measure preserving invertible map. We call this a dynamical system. For a subset A E 5 r, by TA: A --* A we mean the induced map, TA(X ) : TrA(~)(X) where rA(x) = min{i > 0: Ti(x) E A}. Such induced maps can be topologized by the natural metric D(A, A') = tt(A/kA I) on .T" rood sets of measure zero. We discuss here ergodic properties of TA which are residual in this metric. The first theorem is due to Conze. THEOREM 1 (Conze): For T ergodic, TA is weakly mixing for a residual set of A. THEOREM 2: For T ergodic, O-entropy and loosely Bernoulli, TA is rank-1 and rigid for a residual set of A. THEOREM 3: For T ergodic, positive entropy and loosely Bernoulli, TA is Bernoulli for a residual set of A. THEOREM 4: For T ergodic of positive entropy, TA 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 algebra of T. We also discuss even Kakutani equivalence analogues of Theorems 1-4. Received May 22, 1994 387