Skip to main content Accessibility help
×
×
Home

Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour

  • Feliks Przytycki (a1)

Abstract

We find very simple examples of C-arcs of diffeomorphisms of the two-dimensional torus, preserving the Lebesgue measure and having the following properties: (1) the beginning of an arc is inside the set of Anosov diffeomorphisms; (2) after the bifurcation parameter every diffeomorphism has an elliptic fixed point with the first Birkhoff invariant non-zero (the KAM situation) and an invariant open area with almost everywhere non-zero Lyapunov characteristic exponents, moreover where the diffeomorphism has Bernoulli property; (3) the arc is real-analytic except on two circles (for each value of parameter) which are inside the Bernoulli property area.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour
      Available formats
      ×

Copyright

References

Hide All
[1]Burton, R. & Easton, R.. Ergodicity of linked twist mappings. Global Theory of Dynamical Systems: Lectures Notes in Math. No. 819. Springer: Berlin, 1980, pp. 3549.
[2]Edwards, R. E.. Functional Analysis. Holt, Rinehart and Winston: New York, 1965.
[3]Franks, J.. Anosov diffeomorphisms. Global Analysis. Proc. Symp. Pure Math. 14 (1970), 6193.
[4]Gerber, M. & Katok, A.. Smooth models of Thurston's pseudo-Anosov maps. Ann. Sci. de l'Ecole Normale Superieure, Ae serie, 15 (1982), 173204.
[5]Hirsch, M. & Pugh, Ch.. Stable manifolds and hyperbolic sets. Global Analysis, Proc. Symp. Pure Math. 14 (1970), 133163.
[6]Katok, A. Bernoulli diffeomorphisms on surfaces. Ann. Math. 110 (1979), 529547.
[7]Mané, R.. The Lyapunov exponents of generic area preserving diffeomorphisms. Preprint.
[8]Pesin, Ya. B.. Lyapunov characteristic exponents and smooth ergodic theory. Uspekhi Math. Nauk. 32 No. 4 (1977), 55112.
English translation: Ross. Math. Surveys 32 No. 4 (1977), 55114.
[9]Przytycki, F.. Linked twist mappings: Ergodicity. Preprint I.H.E.S., February 1981. (To appear in part in Ann. Scientifiques de I'Ecole Norm. Sup.)
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed