Stretching the Oxtoby-Ulam Theorem

Ethan Akin

ArticleOriginal scientific text

Title

Stretching the Oxtoby-Ulam Theorem

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Affiliations

Mathematics Department, The City College, 137 Street and Convent Avenue, New York, NY 10031, U.S.A.

Abstract

On a manifold X of dimension at least two, let μ be a nonatomic measure of full support with μ(∂X) = 0. The Oxtoby-Ulam Theorem says that ergodicity of μ is a residual property in the group of homeomorphisms which preserve μ. Daalderop and Fokkink have recently shown that density of periodic points is residual as well. We provide a proof of their result which replaces the dependence upon the Annulus Theorem by a direct construction which assures topologically robust periodic points.

E. Akin, The General Topology of Dynamical Systems, Amer. Math. Soc., Providence, 1993.
S. Alpern and V. Prasad, Typical properties of volume preserving homeomorphisms, to appear (2000).
J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems 17 (1997), 505-529.
F. Daalderop and R. Fokkink, Chaotic homeomorphisms are generic, Topology Appl., to appear (2000).
J. Milnor, Topology from the Differentiable Viewpoint, Univ. of Virginia Press, Charlottesville, VA, 1965.
Z. Nitecki and M. Shub, Filtrations, decompositions, and explosions, Amer. J. Math. 97 (1975), 1029-1047.
J. Oxtoby, Note on transitive transformations, Proc. Nat. Acad. Sci. U.S.A. 23 (1937), 443-446.
J. Oxtoby, Diameters of arcs and the gerrymandering problem, Amer. Math. Monthly 84 (1977), 155-162.
J. Oxtoby, Measure and Category, 2nd ed., Springer, New York, NY, 1980.
J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874-920.

Title

Stretching the Oxtoby-Ulam Theorem

Affiliations

Abstract

Bibliography