Measure rigidity for random dynamics on surfaces and related skew products

Brown, Aaron; Hertz, Federico

doi:10.1090/jams/877

Measure rigidity for random dynamics on surfaces and related skew products
HTML articles powered by AMS MathViewer

by Aaron Brown and Federico Rodriguez Hertz

J. Amer. Math. Soc. 30 (2017), 1055-1132

DOI: https://doi.org/10.1090/jams/877

Published electronically: March 7, 2017

PDF | Request permission

Abstract:

Given a surface $M$ and a Borel probability measure $\nu$ on the group of $C^2$-diffeomorphisms of $M$ we study $\nu$-stationary probability measures on $M$. We prove for hyperbolic stationary measures the following trichotomy: the stable distributions are non-random, the measure is SRB, or the measure is supported on a finite set and is hence almost-surely invariant. In the proof of the above results, we study skew products with surface fibers over a measure-preserving transformation equipped with a decreasing sub-$\sigma$-algebra $\hat {\mathcal F}$ and derive a related result. A number of applications of our main theorem are presented.

References

Erik Sparre Andersen and Børge Jessen, Some limit theorems on set-functions, Danske Vid. Selsk. Mat.-Fys. Medd. 25 (1948), no. 5, 8. MR 28376
Artur Avila and Marcelo Viana, Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math. 181 (2010), no. 1, 115–189. MR 2651382, DOI 10.1007/s00222-010-0243-1
L. Backes, A. W. Brown, and C. Butler, Continuity of Lyapunov Exponents for Cocycles with Invariant Holonomies. Preprint, 2015. arXiv:1507.08978.
Jörg Bahnmüller and Thomas Bogenschütz, A Margulis-Ruelle inequality for random dynamical systems, Arch. Math. (Basel) 64 (1995), no. 3, 246–253. MR 1314494, DOI 10.1007/BF01188575
Jörg Bahnmüller and Pei-Dong Liu, Characterization of measures satisfying the Pesin entropy formula for random dynamical systems, J. Dynam. Differential Equations 10 (1998), no. 3, 425–448. MR 1646606, DOI 10.1023/A:1022653229891
Luis Barreira and Yakov Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge, 2007. Dynamics of systems with nonzero Lyapunov exponents. MR 2348606, DOI 10.1017/CBO9781107326026
Luis Barreira, Yakov Pesin, and Jörg Schmeling, Dimension and product structure of hyperbolic measures, Ann. of Math. (2) 149 (1999), no. 3, 755–783. MR 1709302, DOI 10.2307/121072
Yves Benoist and Jean-François Quint, Mesures stationnaires et fermés invariants des espaces homogènes, Ann. of Math. (2) 174 (2011), no. 2, 1111–1162 (French, with English and French summaries). MR 2831114, DOI 10.4007/annals.2011.174.2.8
Yves Benoist and Jean-François Quint, Stationary measures and invariant subsets of homogeneous spaces (II), J. Amer. Math. Soc. 26 (2013), no. 3, 659–734. MR 3037785, DOI 10.1090/S0894-0347-2013-00760-2
Yves Benoist and Jean-François Quint, Stationary measures and invariant subsets of homogeneous spaces (III), Ann. of Math. (2) 178 (2013), no. 3, 1017–1059. MR 3092475, DOI 10.4007/annals.2013.178.3.5
C. Bocker-Neto and M. Viana, Continuity of Lyapunov Exponents for Random 2D Matrices. 2010. arXiv:1012.0872.
Christian Bonatti, Xavier Gómez-Mont, and Marcelo Viana, Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré C Anal. Non Linéaire 20 (2003), no. 4, 579–624 (French, with English and French summaries). MR 1981401, DOI 10.1016/S0294-1449(02)00019-7
Jean Bourgain, Alex Furman, Elon Lindenstrauss, and Shahar Mozes, Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus, J. Amer. Math. Soc. 24 (2011), no. 1, 231–280. MR 2726604, DOI 10.1090/S0894-0347-2010-00674-1
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989, DOI 10.1007/BFb0081279
Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. MR 380889, DOI 10.1007/BF01389848
Aaron W. Brown, Smooth stabilizers for measures on the torus, Discrete Contin. Dyn. Syst. 35 (2015), no. 1, 43–58. MR 3286947, DOI 10.3934/dcds.2015.35.43
Dmitry Dolgopyat and Raphaël Krikorian, On simultaneous linearization of diffeomorphisms of the sphere, Duke Math. J. 136 (2007), no. 3, 475–505. MR 2309172, DOI 10.1215/S0012-7094-07-13633-0
A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL(2,R) action on Moduli space. 2013. arXiv:1302.3320.
Harry Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377–428. MR 163345, DOI 10.1090/S0002-9947-1963-0163345-0
Joseph Horowitz, Optional supermartingales and the Andersen-Jessen theorem, Z. Wahrsch. Verw. Gebiete 43 (1978), no. 3, 263–272. MR 501787, DOI 10.1007/BF00536207
Boris Kalinin and Anatole Katok, Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori, J. Mod. Dyn. 1 (2007), no. 1, 123–146. MR 2261075, DOI 10.3934/jmd.2007.1.123
Boris Kalinin, Anatole Katok, and Federico Rodriguez Hertz, Nonuniform measure rigidity, Ann. of Math. (2) 174 (2011), no. 1, 361–400. MR 2811602, DOI 10.4007/annals.2011.174.1.10
A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 751–778. MR 1406432, DOI 10.1017/S0143385700009081
Yuri Kifer, Ergodic theory of random transformations, Progress in Probability and Statistics, vol. 10, Birkhäuser Boston, Inc., Boston, MA, 1986. MR 884892, DOI 10.1007/978-1-4684-9175-3
F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 163–188 (French). MR 743818, DOI 10.1007/BF02698772
F. Ledrappier, Positivity of the exponent for stationary sequences of matrices, Lyapunov exponents (Bremen, 1984) Lecture Notes in Math., vol. 1186, Springer, Berlin, 1986, pp. 56–73. MR 850070, DOI 10.1007/BFb0076833
F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. MR 819556, DOI 10.2307/1971328
F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2) 122 (1985), no. 3, 540–574. MR 819557, DOI 10.2307/1971329
F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Theory Related Fields 80 (1988), no. 2, 217–240. MR 968818, DOI 10.1007/BF00356103
Pei-Dong Liu and Min Qian, Smooth ergodic theory of random dynamical systems, Lecture Notes in Mathematics, vol. 1606, Springer-Verlag, Berlin, 1995. MR 1369243, DOI 10.1007/BFb0094308
Pei-Dong Liu and Jian-Sheng Xie, Dimension of hyperbolic measures of random diffeomorphisms, Trans. Amer. Math. Soc. 358 (2006), no. 9, 3751–3780. MR 2218998, DOI 10.1090/S0002-9947-06-03933-X
Ricardo Mañé, A proof of Pesin’s formula, Ergodic Theory Dynam. Systems 1 (1981), no. 1, 95–102. MR 627789, DOI 10.1017/s0143385700001188
Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1332–1379, 1440 (Russian). MR 0458490
Min Qian, Min-Ping Qian, and Jian-Sheng Xie, Entropy formula for random dynamical systems: relations between entropy, exponents and dimension, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1907–1931. MR 2032494, DOI 10.1017/S0143385703000178
Min Qian, Jian-Sheng Xie, and Shu Zhu, Smooth ergodic theory for endomorphisms, Lecture Notes in Mathematics, vol. 1978, Springer-Verlag, Berlin, 2009. MR 2542186, DOI 10.1007/978-3-642-01954-8
V. A. Rokhlin. Lectures on the entropy theory of measure-preserving transformations. Russ. Math. Surv. 22 (1967), no. 5, 1–52.
Daniel J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy, Ergodic Theory Dynam. Systems 10 (1990), no. 2, 395–406. MR 1062766, DOI 10.1017/S0143385700005629
Klaus D. Schmidt, The Andersen-Jessen theorem revisited, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 2, 351–361. MR 898154, DOI 10.1017/S0305004100067360
Marcelo Viana, Lectures on Lyapunov exponents, Cambridge Studies in Advanced Mathematics, vol. 145, Cambridge University Press, Cambridge, 2014. MR 3289050, DOI 10.1017/CBO9781139976602