Ergodic Theory and Dynamical Systems

Properties generic for Lebesgue space automorphisms are generic for measure-preserving manifold homeomorphisms

a1 Department of Mathematics, The London School of Economics, Houghton Street, London WC2A 2AE, UK (e-mail:
a2 Department of Mathematics, One University Avenue, University of Massachusetts Lowell, Lowell, MA 01854, USA (e-mail:


First we describe the work of the first author leading to the conclusion that any property generic (in the weak topology) for measure-preserving bijections of a Lebesgue probability space is also generic (in the compact-open topology) for homeomorphisms of a compact manifold preserving a fixed measure. Then we describe the work of both authors in extending this result to non-compact manifolds, with modifications based on the ends of the manifold. These results can be thought of as generalizations of the original work which established genericity for the specific property of ergodicity (Oxtoby and Ulam, 1941) and subsequent work for other properties such as weak mixing (Katok and Stepin, 1970). The techniques used to obtain the titled theorem are also applied to related areas, such as fixed point theorems and chaos theory, and some new results are obtained.

(Received July 1 2000)
(Revised March 7 2002)