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Notes on the multiplicative ergodic theorem

Published online by Cambridge University Press:  07 September 2017

SIMION FILIP
SIMION FILIP*
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02139, USA email [email protected]
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Abstract

The Oseledets multiplicative ergodic theorem is a basic result with numerous applications throughout dynamical systems. These notes provide an introduction to this theorem, as well as subsequent generalizations. They are based on lectures at summer schools in Brazil, France, and Russia.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1153 - 1189
Copyright
© Cambridge University Press, 2017 

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References

Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50) . American Mathematical Society, Providence, RI, 1997.Google Scholar
Arveson, W.. An Invitation to C ∗ -Algebras (Graduate Texts in Mathematics, 39) . Springer, New York, 1976.Google Scholar
Bridson, M. R. and Haefliger, A.. Metric Spaces of Non-positive Curvature (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319) . Springer, Berlin, 1999.Google Scholar
Bochi, J.. The multiplicative ergodic theorem of Oseledets. Lecture notes available on the author’s website, http://www.mat.uc.cl/∼jairo.bochi/docs/oseledets.pdf.Google Scholar
Bogachev, V. I.. Measure Theory. Vols. I, II. Springer, Berlin, 2007.Google Scholar
Einsiedler, M. and Ward, T.. Ergodic Theory with a View Towards Number Theory (Graduate Texts in Mathematics, 259) . Springer, London, 2011.Google Scholar
Furstenberg, H. and Kesten, H.. Products of random matrices. Ann. Math. Statist. 31 (1960), 457–469.Google Scholar
Gouëzel, S. and Karlsson, A.. Subadditive and multiplicative ergodic theorems. ArXiv e-prints, Preprint, 2015, arXiv:1509.07733 [math.DS].Google Scholar
Kaĭmanovich, V. A.. Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) [Differentsialnaya Geom. Gruppy Li i Mekh. IX] 164 (1987), 29–46, 196–197.Google Scholar
Karlsson, A.. Ergodic theorems for noncommuting random products. Lecture notes available on the author’s website, http://www.unige.ch/math/folks/karlsson/wroclawtotal.pdf.Google Scholar
Kechris, A. S.. Classical Descriptive Set Theory (Graduate Texts in Mathematics, 156) . Springer, New York, 1995.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54) . Cambridge University Press, Cambridge, 1995, with a supplementary chapter by Katok and Leonardo Mendoza.Google Scholar
Karlsson, A. and Ledrappier, F.. On laws of large numbers for random walks. Ann. Probab. 34(5) (2006), 1693–1706.Google Scholar
Karlsson, A. and Margulis, G. A.. A multiplicative ergodic theorem and nonpositively curved spaces. Comm. Math. Phys. 208(1) (1999), 107–123.Google Scholar
Ledrappier, F.. Quelques propriétés des exposants caractéristiques. École d’été de probabilités de Saint-Flour, XII—1982 (Lecture Notes in Mathematics, 1097) . Springer, Berlin, 1984, pp. 305–396.Google Scholar
Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann. of Math. (2) 122(3) (1985), 509–539.Google Scholar
Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. (2) 122(3) (1985), 540–574.Google Scholar
Mañé, R.. Ergodic Theory and Differentiable Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8) . Springer, Berlin, 1987, translated from the Portuguese by Silvio Levy.Google Scholar
Margulis, G. A.. Arithmetic properties of discrete subgroups. Uspekhi Mat. Nauk 29(1(175)) (1974), 49–98.Google Scholar
Monod, N.. Superrigidity for irreducible lattices and geometric splitting. J. Amer. Math. Soc. 19(4) (2006), 781–814.Google Scholar
Morris, I. D.. Mather sets for sequences of matrices and applications to the study of joint spectral radii. Proc. Lond. Math. Soc. (3) 107(1) (2013), 121–150.Google Scholar
Oseledets, V. I.. A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. Tr. Mosk. Mat. Obš. 19 (1968), 179–210.Google Scholar
Pesin, J. B.. Families of invariant manifolds that correspond to nonzero characteristic exponents. Izv. Akad. Nauk SSSR Ser. Mat. 40(6) (1976), 1332–1379, 1440.Google Scholar
Pesin, J. B.. Characteristic Ljapunov exponents, and smooth ergodic theory. Uspekhi Mat. Nauk 32(4(196)) (1977), 55–112 287.Google Scholar
Ruelle, D.. Ergodic theory of differentiable dynamical systems. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27–58.Google Scholar
Ruelle, D.. Characteristic exponents and invariant manifolds in Hilbert space. Ann. of Math. (2) 115(2) (1982), 243–290.Google Scholar
Viana, M.. Lectures on Lyapunov Exponents (Cambridge Studies in Advanced Mathematics, 145) . Cambridge University Press, Cambridge, 2014.Google Scholar
Walters, P.. Ergodic Theory—Introductory Lectures (Lecture Notes in Mathematics, 458) . Springer, Berlin–New York, 1975.Google Scholar
Walters, P.. A dynamical proof of the multiplicative ergodic theorem. Trans. Amer. Math. Soc. 335(1) (1993), 245–257.Google Scholar
Zimmer, R. J.. Ergodic Theory and Semisimple Groups (Monographs in Mathematics, 81) . Birkhäuser, Basel, 1984.Google Scholar