Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T03:20:33.367Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffeomorphisms of the k-torus with wandering domains

Published online by Cambridge University Press:  14 October 2010

Patrick D. McSwiggen
Patrick D. McSwiggen
Affiliation:
Department of Mathematical Sciences, The University of Cincinnati, Cincinnati OH 45221–0025, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that diffeomorphisms analogous to a classical example on the circle due to Denjoy can be constructed on the general k-torus. Such a diffeomorphism has the property that it is semiconjugate to an ergodic translation but has a wandering domain with dense orbit. The construction on the k-torus can be made Cr, and by a Cr small perturbation of a translation, for any r < k + 1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 6 , December 1995 , pp. 1189 - 1205
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Denjoy, A.. Sur les courbes définies par les équations différentielles à la surface du tore. J. Math. Pure Appl. 11 (9) (1932), 333–375.Google Scholar
[2]Hall, G. R.. Bifurcation of an attracting invariant circle: a Denjoy attractor. Ergod. Th. & Dynam. Sys. 3 (1983), 87–118.CrossRefGoogle Scholar
[3]Harrison, J.. Unsmoothable diffeomorphisms. Ann. Math. 102 (1975), 83–94.CrossRefGoogle Scholar
[4]Harrison, J.. Unsmoothable diffeomorphisms on higher dimensional manifolds. Proc. Amer. Math. Soc. 73 (1979), 249–255.CrossRefGoogle Scholar
[5]Herman, M.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. IHES 49 (1979), 1–233.CrossRefGoogle Scholar
[6]Herman, M.. Private communication.Google Scholar
[7]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds, Springer Lecture Notes in Mathematics 583. Springer, New York, 1977.CrossRefGoogle Scholar
[8]McSwiggen, P.. Diffeomorphisms of the torus with wandering domains. Proc. Amer. Math. Soc. 117 (1993), 1175–1186.CrossRefGoogle Scholar
[9]Palis, J. and de Melo, W.. Geometric Theory of Dynamical Systems. Springer, New York, 1982.CrossRefGoogle Scholar
[10]Schmidt, W.. Diophantine Approximation. Springer Lecture Notes in Mathematics 785. Springer, New York, 1980.Google Scholar
[11]Sullivan, D.. Itération des fonctions analytique complexes. C R Acad. Sci. Paris, Sér. I Math. 294 (1982), 301–304.Google Scholar
[12]Walters, P.. Anosov diffeomorphisms are topologically stable. Topology 9 (1970), 71–78.CrossRefGoogle Scholar
[13]Williams, R. F.. The ‘DA’ maps of Smale and structural stability. Global Analysis, Proc. Symp. Pure Math. 14 (1970), 328–334.Google Scholar