Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-17T15:43:03.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimensional characteristics of invariant measures for circle diffeomorphisms

Published online by Cambridge University Press:  03 February 2009

VICTORIA SADOVSKAYA
VICTORIA SADOVSKAYA*
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA (email: sadovska@jaguar1.usouthal.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number τ there exists a C∞ circle diffeomorphism with rotation number τ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value 0≤β≤1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1979 - 1992
Copyright
Copyright © Cambridge University Press 2009

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

[1]Anosov, D. and Katok, A.. New examples in smooth ergodic theory. Ergodic diffeomorphisms. Trans. Moscow Math. Soc. 23 (1970), 1–35.Google Scholar
[2]Barreira, L., Pesin, Ya. and Schmeling, J.. Dimension and product structure of hyperbolic measures. Ann. of Math. (2) 149(3) (1999), 755–783.CrossRefGoogle Scholar
[3]Eckmann, J. P. and Ruelle, D.. Ergodic theory of chaos and strange attractors. Rev. Modern Phys. 57(3) (1985), 617–656.CrossRefGoogle Scholar
[4]Fayad, B. and Saprykina, M.. Weak mixing disc and annulus diffeomorphisms with arbitrary Liouvillean rotation number on the boundary. Ann. Sci. École Norm. Sup. 38(3) (2005), 339–364.CrossRefGoogle Scholar
[5]Fayad, B., Saprykina, M. and Windsor, A.. Non-standard smooth realizations of Liouville rotations. Ergod. Th. & Dynam. Sys. 27 (2007), 1803–1818.CrossRefGoogle Scholar
[6]Herman, M.-R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5–233.CrossRefGoogle Scholar
[7]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, London, 1995.CrossRefGoogle Scholar
[8]Kalinin, B. and Sadovskaya, V.. On pointwise dimension of nonhyperbolic measures. Ergod. Th. & Dynam. Sys. 22(6) (2002), 1783–1801.CrossRefGoogle Scholar
[9]Ledrappier, F. and Misiurewicz, M.. Dimension of invariant measures for maps with exponent zero. Ergod. Th. & Dynam. Sys. 5 (1985), 595–610.CrossRefGoogle Scholar
[10]Pesin, Ya.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications. The University of Chicago Press, Chicago, IL, 1998.Google Scholar
[11]Young, L.-S.. Dimension, entropy, and lyapunov exponents. Ergod. Th. & Dynam. Sys. 2 (1982), 109–124.CrossRefGoogle Scholar