American Institute of Mathematical Sciences

Advanced
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), 339-364. 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), 5-233.  Google Scholar

[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.  Google Scholar

[4]

V. Sadovskaya, Dimensional characteristics of invariant measures for circle diffeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1979-1992. 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), 333-359.  Google Scholar

[6]

J.-C. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle, Astérisque, 231 (1995), 89-242.  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), 339-364. 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), 5-233.  Google Scholar

[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.  Google Scholar

[4]

V. Sadovskaya, Dimensional characteristics of invariant measures for circle diffeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1979-1992. 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), 333-359.  Google Scholar

[6]

J.-C. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle, Astérisque, 231 (1995), 89-242.  Google Scholar

[1]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[2]

Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007
[3]

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335
[4]

Wenxian Shen. Global attractor and rotation number of a class of nonlinear noisy oscillators. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 597-611. doi: 10.3934/dcds.2007.18.597
[5]

Hicham Zmarrou, Ale Jan Homburg. Dynamics and bifurcations of random circle diffeomorphism. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 719-731. doi: 10.3934/dcdsb.2008.10.719
[6]

Laurent Imbert, Michael J. Jacobson, Jr., Arthur Schmidt. Fast ideal cubing in imaginary quadratic number and function fields. Advances in Mathematics of Communications, 2010, 4 (2) : 237-260. doi: 10.3934/amc.2010.4.237
[7]

Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015
[8]

Harald Fripertinger. The number of invariant subspaces under a linear operator on finite vector spaces. Advances in Mathematics of Communications, 2011, 5 (2) : 407-416. doi: 10.3934/amc.2011.5.407
[9]

Qiudong Wang. The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 255-274. doi: 10.3934/dcds.2000.6.255
[10]

Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729
[11]

Romar dela Cruz, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. On the minimum number of minimal codewords. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020130
[12]

Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377
[13]

Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647
[14]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365
[15]

Wilfrid Gangbo, Andrzej Świech. Optimal transport and large number of particles. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1397-1441. doi: 10.3934/dcds.2014.34.1397
[16]

Sujay Jayakar, Robert S. Strichartz. Average number of lattice points in a disk. Communications on Pure & Applied Analysis, 2016, 15 (1) : 1-8. doi: 10.3934/cpaa.2016.15.1
[17]

G.F. Webb. The prime number periodical cicada problem. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 387-399. doi: 10.3934/dcdsb.2001.1.387
[18]

Abdelhamid Adouani, Habib Marzougui. Computation of rotation numbers for a class of PL-circle homeomorphisms. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3399-3419. doi: 10.3934/dcds.2012.32.3399
[19]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591
[20]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences