Stretching the Oxtoby-Ulam Theorem

Ethan Akin

Colloquium Mathematicae (2000)

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

Abstract

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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.

How to cite

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Akin, 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

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