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

Smooth stabilizers for measures on the torus

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.
Citation: Aaron W. Brown. Smooth stabilizers for measures on the torus. Discrete and 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.

[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.

[3]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Mathematics and its Applications, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.

[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.

[5]

A. W. Brown, Rigidity Properties of Measures on the Torus, Ph.D thesis, Tufts University, 2011.

[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.

[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.

[8]

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

[9]

H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100. doi: 10.1017/S0143385700007215.

[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.

[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.

[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.

[13]

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

[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.

[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.

[16]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: 10.2307/2373551.

[17]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. doi: 10.2307/2373372.

[18]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.

[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.

[20]

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

[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.

[22]

V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55 pp.

[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.

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.

[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.

[3]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Mathematics and its Applications, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.

[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.

[5]

A. W. Brown, Rigidity Properties of Measures on the Torus, Ph.D thesis, Tufts University, 2011.

[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.

[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.

[8]

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

[9]

H. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100. doi: 10.1017/S0143385700007215.

[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.

[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.

[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.

[13]

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

[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.

[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.

[16]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: 10.2307/2373551.

[17]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. doi: 10.2307/2373372.

[18]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.

[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.

[20]

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

[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.

[22]

V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55 pp.

[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.

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