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