American Institute of Mathematical Sciences

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 & 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

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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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