# Stretching the Oxtoby-Ulam Theorem

Colloquium Mathematicae (2000)

- Volume: 84/85, Issue: 1, page 83-94
- ISSN: 0010-1354

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topAkin, Ethan. "Stretching the Oxtoby-Ulam Theorem." Colloquium Mathematicae 84/85.1 (2000): 83-94. <http://eudml.org/doc/210811>.

@article{Akin2000,

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.},

author = {Akin, Ethan},

journal = {Colloquium Mathematicae},

keywords = {Oxtoby-Ulam theorem; weak mixing; periodic point},

language = {eng},

number = {1},

pages = {83-94},

title = {Stretching the Oxtoby-Ulam Theorem},

url = {http://eudml.org/doc/210811},

volume = {84/85},

year = {2000},

}

TY - JOUR

AU - Akin, Ethan

TI - Stretching the Oxtoby-Ulam Theorem

JO - Colloquium Mathematicae

PY - 2000

VL - 84/85

IS - 1

SP - 83

EP - 94

AB - 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.

LA - eng

KW - Oxtoby-Ulam theorem; weak mixing; periodic point

UR - http://eudml.org/doc/210811

ER -

## References

top- [1] E. Akin, The General Topology of Dynamical Systems, Amer. Math. Soc., Providence, 1993. Zbl0781.54025
- [2] S. Alpern and V. Prasad, Typical properties of volume preserving homeomorphisms, to appear (2000). Zbl0970.37001
- [3] J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems 17 (1997), 505-529. Zbl0921.54029
- [4] F. Daalderop and R. Fokkink, Chaotic homeomorphisms are generic, Topology Appl., to appear (2000). Zbl0977.54032
- [5] J. Milnor, Topology from the Differentiable Viewpoint, Univ. of Virginia Press, Charlottesville, VA, 1965.
- [6] Z. Nitecki and M. Shub, Filtrations, decompositions, and explosions, Amer. J. Math. 97 (1975), 1029-1047. Zbl0324.58015
- [7] J. Oxtoby, Note on transitive transformations, Proc. Nat. Acad. Sci. U.S.A. 23 (1937), 443-446. Zbl0017.13603
- [8] J. Oxtoby, Diameters of arcs and the gerrymandering problem, Amer. Math. Monthly 84 (1977), 155-162. Zbl0355.52007
- [9] J. Oxtoby, Measure and Category, 2nd ed., Springer, New York, NY, 1980. Zbl0435.28011
- [10] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874-920. Zbl0063.06074

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