Ergodic Theory and Area Preserving Mappings

  • Robert W. Easton
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 82)

Abstract

Much work has been done recently in numerical (computer) studies of area preserving mappings of the plane. Often the orbit of a single point seems to fill a region in the plane having positive area. Such a region is called an “ergodic zone”. In this paper we give an exposition of some mathematical techniques which can be used to show that under suitable hypotheses, the closure of an orbit of a single point has positive Lebesque measure. We then apply these techniques to show that a family of mappings called linked twist maps are ergodic.

Keywords

Manifold Peri Boulder 

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References

  1. Arnold, V., (1978), Math. Methods in Classical Mech., Graduate Texts in Math #60, Springer-Verlag.Google Scholar
  2. Burton R. and R. Easton, (1979), Ergodicity of Linked Twist Maps, Lecture Notes in Math. 819, Springer-Verlag.Google Scholar
  3. Billingsley, P., (1965), Ergodic Theory and Information, John Wiley and Sons Inc.Google Scholar
  4. Easton, R. (1979), Perturbed Twist Maps, Homoclinic Points and Ergodic Zones, Instabilities in Dynamical Systems, (ed. V. Szebehely ), D. Reidel Publ. Co.Google Scholar
  5. Easton, R., Chain Transitivity and the Domain of Influence of an Invariant Set, Lect. Notes in Math. No. 668, Springer-Verlag.Google Scholar
  6. Devaney, D., (1979), Linked Twist Maps are Almost Anosov, Lect. Notes in Math. 819, Springer-Verlag.Google Scholar
  7. Herman, M., (1981), to appear.Google Scholar
  8. Katok, A., Y. Sinai, A. Stepin, (1977) Theory of Dynamical Systems and General Transformation Groups with an Invariant Measure, J. of Soviet Math., 7, pp. 974 - 1065.CrossRefGoogle Scholar
  9. Moser, J., (1973), Stable and Random Motions in Dynamical Systems, Annals of Math. Study 77, Princeton Univ. Press.Google Scholar
  10. Pesin, R., (1977), Lyapunov Characteristic Exponents and Smooth Ergodic Theory, Russian Math. Surveys, 32, pp. 55 – 114.MathSciNetCrossRefGoogle Scholar
  11. Przytycki, F., (1981), Linked Twist Mappings: Ergodicity, IHES Pub. M/81/20.Google Scholar
  12. Ruelle, D., (1978), Ergodic Theory of Differentiable Dynamical Systems, IHES. Pub. P/78/240.Google Scholar
  13. Simo, C., (1981) See this volume, p. 357.Google Scholar
  14. Weiss, B., (1975), The Geodesic Flow on Surfaces of Negative Curvature, Lect. Notes in Physics No. 38, Springer-Verlag.Google Scholar
  15. Wojtkowski, M., (1979), Linked Twist Mappings have the K-Property, In Nonlinear Dynamics, Annals N. Y. Academy of Sciences, No. 357 (ed. R. Helleman).Google Scholar

Copyright information

© D. Reidel Publishing Company 1982

Authors and Affiliations

  • Robert W. Easton
    • 1
  1. 1.University of ColoradoBoulderUSA

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