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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, 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, Providence, RI, 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), 1549-1573. doi: 10.1088/0305-4470/30/5/020.  Google Scholar

[3]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Mathematics and its Applications, Cambridge University Press, Cambridge, 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), 755-783. doi: 10.2307/121072.  Google Scholar

[5]

A. W. Brown, Rigidity Properties of Measures on the Torus, Ph.D thesis, Tufts University, 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), 99-110. doi: 10.1090/S1079-6762-03-00117-3.  Google Scholar

[7]

J. Franks, Anosov diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 61-93.  Google Scholar

[8]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems, 14 (1994), 645-666. doi: 10.1017/S0143385700008105.  Google Scholar

[9]

H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100. 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), 123-146.  Google Scholar

[11]

B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity, Ann. of Math. (2), 174 (2011), 361-400. 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), 751-778. doi: 10.1017/S0143385700009081.  Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 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), 540-574. doi: 10.2307/1971329.  Google Scholar

[15]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 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), 422-429. doi: 10.2307/2373551.  Google Scholar

[17]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. 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), 179-210.  Google Scholar

[19]

J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Annals of Mathematics. Second Series, 42 (1941), 874-920. doi: 10.2307/1968772.  Google Scholar

[20]

J. Palis, C. Pugh and R. C. Robinson, Nondifferentiability of invariant foliations, in Dynamical Systems-Warwick 1974, Springer, (1975), 234-240.  Google Scholar

[21]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546. 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), 55 pp.  Google Scholar

[23]

D. J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy, Ergodic Theory Dynam. Systems, 10 (1990), 395-406. 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, Providence, RI, 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), 1549-1573. doi: 10.1088/0305-4470/30/5/020.  Google Scholar

[3]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Mathematics and its Applications, Cambridge University Press, Cambridge, 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), 755-783. doi: 10.2307/121072.  Google Scholar

[5]

A. W. Brown, Rigidity Properties of Measures on the Torus, Ph.D thesis, Tufts University, 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), 99-110. doi: 10.1090/S1079-6762-03-00117-3.  Google Scholar

[7]

J. Franks, Anosov diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 61-93.  Google Scholar

[8]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems, 14 (1994), 645-666. doi: 10.1017/S0143385700008105.  Google Scholar

[9]

H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100. 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), 123-146.  Google Scholar

[11]

B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity, Ann. of Math. (2), 174 (2011), 361-400. 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), 751-778. doi: 10.1017/S0143385700009081.  Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 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), 540-574. doi: 10.2307/1971329.  Google Scholar

[15]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 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), 422-429. doi: 10.2307/2373551.  Google Scholar

[17]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. 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), 179-210.  Google Scholar

[19]

J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Annals of Mathematics. Second Series, 42 (1941), 874-920. doi: 10.2307/1968772.  Google Scholar

[20]

J. Palis, C. Pugh and R. C. Robinson, Nondifferentiability of invariant foliations, in Dynamical Systems-Warwick 1974, Springer, (1975), 234-240.  Google Scholar

[21]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546. 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), 55 pp.  Google Scholar

[23]

D. J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy, Ergodic Theory Dynam. Systems, 10 (1990), 395-406. doi: 10.1017/S0143385700005629.  Google Scholar

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