A generic-dimensional propertyof the invariant measures for circle diffeomorphisms

Shigenori Matsumoto

doi:10.3934/jmd.2013.7.553

Article Contents

2013, Volume 7, Issue 4: 553-563. Doi: 10.3934/jmd.2013.7.553

This issue Previous Article Entropic stability beyond partial hyperbolicity Next Article Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation

A generic-dimensional property of the invariant measures for circle diffeomorphisms

Shigenori Matsumoto^1,

1.
Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda, Surugadai, Chiyoda-ku, Tokyo, 101-8308

Received: November 30, 2012

Revised: November 30, 2013

Published: February 2014

Abstract Related Papers Cited by

Abstract

Given any Liouville number $\alpha$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation-preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.

Keywords:

Mathematics Subject Classification: Primary: 37E10; Secondary: 37E45.

Citation:

Related Papers

Cited by

References

[1]	B. Fayad and M. Saprykina, Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary, Ann. Sci. École Norn. Sup. (4), 38 (2005), 339-364.doi: 10.1016/j.ansens.2005.03.004.
[2]	M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233.
[3]	S. Matsumoto, Dense properties of the space of circle diffeomorphisms with a Liouville rotation number, Nonlinearity, 25 (2012), 1495-1511.doi: 10.1088/0951-7715/25/5/1495.
[4]	V. Sadovskaya, Dimensional characteristics of invariant measures for circle diffeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1979-1992.doi: 10.1017/S0143385708000916.
[5]	J.-C. Yoccoz, Conjugaison différentiable des difféimorphismes du cercle dont le nombre de rotation vérifie une conditon diophantienne, Ann. Sci. Ecole Norm. Sup. (4), 17 (1984), 333-359.
[6]	J.-C. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle, Astérisque, 231 (1995), 89-242.