American Institute of Mathematical Sciences

Advanced
January  2015, 35(1): 43-58. doi: 10.3934/dcds.2015.35.43

Smooth stabilizers for measures on the torus

Aaron W. Brown 1,

1. 

Department of Mathematics, The Pennsylvania State University, State College, PA 16802, United States

Received  February 2013 Revised  June 2014 Published  August 2014

For a dissipative Anosov diffeomorphism $f$ of the 2-torus, we give examples of $f$-invariant measures $\mu$ such that the group of $\mu$-preserving diffeomorphisms is virtually cyclic.
Keywords: hyperbolic dynamics., rigidity, invariant measures, Anosov diffeomorphisms, group actions.
Mathematics Subject Classification: Primary: 37C40, 37D20; Secondary: 37E3.
Citation: Aaron W. Brown. Smooth stabilizers for measures on the torus. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 43-58. doi: 10.3934/dcds.2015.35.43
References:
[1]

S. K. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces,, American Mathematical Society, (1996).   Google Scholar

[2]

M. Baake and J. A. G. Roberts, Reversing symmetry group of $ Gl(2,\mathbbZ)$ and $ PGl(2,\mathbbZ)$ matrices with connections to cat maps and trace maps,, J. Phys. A, 30 (1997), 1549.  doi: 10.1088/0305-4470/30/5/020.  Google Scholar

[3]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity,, Mathematics and its Applications, (2007).  doi: 10.1017/CBO9781107326026.  Google Scholar

[4]

L. Barreira, Y. Pesin and J. Schmeling, Dimension and product structure of hyperbolic measures,, Ann. of Math. (2), 149 (1999), 755.  doi: 10.2307/121072.  Google Scholar

[5]

A. W. Brown, Rigidity Properties of Measures on the Torus,, Ph.D thesis, (2011).   Google Scholar

[6]

M. Einsiedler and E. Lindenstrauss, Rigidity properties of $\mathbbZ^d$-actions on tori and solenoids,, Electron. Res. Announc. Amer. Math. Soc., 9 (2003), 99.  doi: 10.1090/S1079-6762-03-00117-3.  Google Scholar

[7]

J. Franks, Anosov diffeomorphisms,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 61.   Google Scholar

[8]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations,, Ergodic Theory Dynam. Systems, 14 (1994), 645.  doi: 10.1017/S0143385700008105.  Google Scholar

[9]

H. Hu, Some ergodic properties of commuting diffeomorphisms,, Ergodic Theory Dynam. Systems, 13 (1993), 73.  doi: 10.1017/S0143385700007215.  Google Scholar

[10]

B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori,, J. Mod. Dyn., 1 (2007), 123.   Google Scholar

[11]

B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity,, Ann. of Math. (2), 174 (2011), 361.  doi: 10.4007/annals.2011.174.1.10.  Google Scholar

[12]

A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions,, Ergodic Theory Dynam. Systems, 16 (1996), 751.  doi: 10.1017/S0143385700009081.  Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar

[14]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension,, Ann. of Math. (2), 122 (1985), 540.  doi: 10.2307/1971329.  Google Scholar

[15]

R. Mañé, Ergodic Theory and Differentiable Dynamics,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), (1987).  doi: 10.1007/978-3-642-70335-5.  Google Scholar

[16]

A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422.  doi: 10.2307/2373551.  Google Scholar

[17]

S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math., 92 (1970), 761.  doi: 10.2307/2373372.  Google Scholar

[18]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems,, Trudy Moskov. Mat. Obšč., 19 (1968), 179.   Google Scholar

[19]

J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Annals of Mathematics. Second Series, 42 (1941), 874.  doi: 10.2307/1968772.  Google Scholar

[20]

J. Palis, C. Pugh and R. C. Robinson, Nondifferentiability of invariant foliations,, in Dynamical Systems-Warwick 1974, (1975), 234.   Google Scholar

[21]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations,, Duke Math. J., 86 (1997), 517.  doi: 10.1215/S0012-7094-97-08616-6.  Google Scholar

[22]

V. A. Rohlin, On the fundamental ideas of measure theory,, Amer. Math. Soc. Translation, 1952 (1952).   Google Scholar

[23]

D. J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy,, Ergodic Theory Dynam. Systems, 10 (1990), 395.  doi: 10.1017/S0143385700005629.  Google Scholar

show all references

References:
[1]

S. K. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces,, American Mathematical Society, (1996).   Google Scholar

[2]

M. Baake and J. A. G. Roberts, Reversing symmetry group of $ Gl(2,\mathbbZ)$ and $ PGl(2,\mathbbZ)$ matrices with connections to cat maps and trace maps,, J. Phys. A, 30 (1997), 1549.  doi: 10.1088/0305-4470/30/5/020.  Google Scholar

[3]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity,, Mathematics and its Applications, (2007).  doi: 10.1017/CBO9781107326026.  Google Scholar

[4]

L. Barreira, Y. Pesin and J. Schmeling, Dimension and product structure of hyperbolic measures,, Ann. of Math. (2), 149 (1999), 755.  doi: 10.2307/121072.  Google Scholar

[5]

A. W. Brown, Rigidity Properties of Measures on the Torus,, Ph.D thesis, (2011).   Google Scholar

[6]

M. Einsiedler and E. Lindenstrauss, Rigidity properties of $\mathbbZ^d$-actions on tori and solenoids,, Electron. Res. Announc. Amer. Math. Soc., 9 (2003), 99.  doi: 10.1090/S1079-6762-03-00117-3.  Google Scholar

[7]

J. Franks, Anosov diffeomorphisms,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 61.   Google Scholar

[8]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations,, Ergodic Theory Dynam. Systems, 14 (1994), 645.  doi: 10.1017/S0143385700008105.  Google Scholar

[9]

H. Hu, Some ergodic properties of commuting diffeomorphisms,, Ergodic Theory Dynam. Systems, 13 (1993), 73.  doi: 10.1017/S0143385700007215.  Google Scholar

[10]

B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori,, J. Mod. Dyn., 1 (2007), 123.   Google Scholar

[11]

B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity,, Ann. of Math. (2), 174 (2011), 361.  doi: 10.4007/annals.2011.174.1.10.  Google Scholar

[12]

A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions,, Ergodic Theory Dynam. Systems, 16 (1996), 751.  doi: 10.1017/S0143385700009081.  Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar

[14]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension,, Ann. of Math. (2), 122 (1985), 540.  doi: 10.2307/1971329.  Google Scholar

[15]

R. Mañé, Ergodic Theory and Differentiable Dynamics,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), (1987).  doi: 10.1007/978-3-642-70335-5.  Google Scholar

[16]

A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422.  doi: 10.2307/2373551.  Google Scholar

[17]

S. E. Newhouse, On codimension one Anosov diffeomorphisms,, Amer. J. Math., 92 (1970), 761.  doi: 10.2307/2373372.  Google Scholar

[18]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems,, Trudy Moskov. Mat. Obšč., 19 (1968), 179.   Google Scholar

[19]

J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Annals of Mathematics. Second Series, 42 (1941), 874.  doi: 10.2307/1968772.  Google Scholar

[20]

J. Palis, C. Pugh and R. C. Robinson, Nondifferentiability of invariant foliations,, in Dynamical Systems-Warwick 1974, (1975), 234.   Google Scholar

[21]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations,, Duke Math. J., 86 (1997), 517.  doi: 10.1215/S0012-7094-97-08616-6.  Google Scholar

[22]

V. A. Rohlin, On the fundamental ideas of measure theory,, Amer. Math. Soc. Translation, 1952 (1952).   Google Scholar

[23]

D. J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy,, Ergodic Theory Dynam. Systems, 10 (1990), 395.  doi: 10.1017/S0143385700005629.  Google Scholar

[1]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123
[2]

Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz. Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data". Journal of Modern Dynamics, 2010, 4 (1) : 207-209. doi: 10.3934/jmd.2010.4.207
[3]

Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068
[4]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271
[5]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68
[6]

A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

[7]

Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193
[8]

Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811
[9]

João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837
[10]

Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 177-202. doi: 10.3934/dcds.2001.7.177
[11]

Lennard F. Bakker, Pedro Martins Rodrigues. A profinite group invariant for hyperbolic toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1965-1976. doi: 10.3934/dcds.2012.32.1965
[12]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[13]

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
[14]

Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80
[15]

Stefano Galatolo, Mathieu Hoyrup, Cristóbal Rojas. Dynamics and abstract computability: Computing invariant measures. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 193-212. doi: 10.3934/dcds.2011.29.193
[16]

Zhenqi Jenny Wang. New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions. Journal of Modern Dynamics, 2010, 4 (4) : 585-608. doi: 10.3934/jmd.2010.4.585
[17]

Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487
[18]

Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985
[19]

Danijela Damjanović. Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions. Journal of Modern Dynamics, 2007, 1 (4) : 665-688. doi: 10.3934/jmd.2007.1.665
[20]

Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences