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October  2013, 7(4): 553-563. doi: 10.3934/jmd.2013.7.553

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, Japan

Received  December 2012 Revised  December 2013 Published  March 2014

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: Hausdorff dimension, fast approximation by conjugation., invariant measure, rotation number, Liouville number, Circle diffeomorphism.
Mathematics Subject Classification: Primary: 37E10; Secondary: 37E4.
Citation: Shigenori Matsumoto. A generic-dimensional property of the invariant measures for circle diffeomorphisms. Journal of Modern Dynamics, 2013, 7 (4) : 553-563. doi: 10.3934/jmd.2013.7.553
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.  doi: 10.1016/j.ansens.2005.03.004.  Google Scholar

[2]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5.   Google Scholar

[3]

S. Matsumoto, Dense properties of the space of circle diffeomorphisms with a Liouville rotation number,, Nonlinearity, 25 (2012), 1495.  doi: 10.1088/0951-7715/25/5/1495.  Google Scholar

[4]

V. Sadovskaya, Dimensional characteristics of invariant measures for circle diffeomorphisms,, Ergodic Theory Dynam. Systems, 29 (2009), 1979.  doi: 10.1017/S0143385708000916.  Google Scholar

[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.   Google Scholar

[6]

J.-C. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle,, Astérisque, 231 (1995), 89.   Google Scholar

show all references

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.  doi: 10.1016/j.ansens.2005.03.004.  Google Scholar

[2]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5.   Google Scholar

[3]

S. Matsumoto, Dense properties of the space of circle diffeomorphisms with a Liouville rotation number,, Nonlinearity, 25 (2012), 1495.  doi: 10.1088/0951-7715/25/5/1495.  Google Scholar

[4]

V. Sadovskaya, Dimensional characteristics of invariant measures for circle diffeomorphisms,, Ergodic Theory Dynam. Systems, 29 (2009), 1979.  doi: 10.1017/S0143385708000916.  Google Scholar

[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.   Google Scholar

[6]

J.-C. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle,, Astérisque, 231 (1995), 89.   Google Scholar

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