Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2000-06

May 2000

Extensions of the Oxtoby-Ulam Theorem on the prevalence of ergodicity for measure preserving homeomorphisms

Steve Alpern and V.S. Prasad


While they were Junior Fellows at Harvard in the 1930's, J.C. Oxtoby and S.M. Ulam worked on the conjecture of G.D. Birkhoff and E. Hopf that ergodicity is the `general case' for homeomorphisms of a manifold preserving a fixed measure. Under the guidance of M. Stone, they proved this conjecture in an important paper published in 1941. Much later A. Katok and A. Stepin extended this result to the property of weak mixing, and afterwards the first author gave combinatorial proofs of both results using a technique of P. Lax.

This article surveys the subsequent work of the authors in extending the result of Oxtoby and Ulam in two different directions. First we describe the work of the first author leading to the conclusion that any property generic for measure preserving bijections of a Lebesgue probability space is also generic for homeomorphisms of a compact manifold preserving a fixed measure. Then we describe the work of both authors in extending the work of Oxtoby and Ulam to noncompact manifolds, with modifications based on the ends of the manifold

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