Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Lectures on Lyapunov Exponents
  • Cited by 106
Publisher:
Cambridge University Press
Online publication date:
July 2014
Print publication year:
2014
Online ISBN:
9781139976602
DOI:
https://doi.org/10.1017/CBO9781139976602
Subjects:
Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Abstract Analysis
Series:
Cambridge Studies in Advanced Mathematics (145)
Information

Book description

The theory of Lyapunov exponents originated over a century ago in the study of the stability of solutions of differential equations. Written by one of the subject's leading authorities, this book is both an account of the classical theory, from a modern view, and an introduction to the significant developments relating the subject to dynamical systems, ergodic theory, mathematical physics and probability. It is based on the author's own graduate course and is reasonably self-contained with an extensive set of exercises provided at the end of each chapter. This book makes a welcome addition to the literature, serving as a graduate text and a valuable reference for researchers in the field.

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
References
[1] P., Anderson. Absence of diffusion in certain random lattices. Phys. Rev., 109:1492–1505, 1958.
[2] D. V., Anosov. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Math. Inst., 90:1–235, 1967.
[3] V., Araújo and M., Bessa. Dominated splitting and zero volume for incompressible three-flows. Nonlinearity, 21:1637–1653, 2008.
[4] A., Arbieto and J., Bochi. Lp-generic cocycles have one-point Lyapunov spectrum. Stoch. Dyn., 3:73–81, 2003.
[5] L., Arnold. Random Dynamical Systems. Springer-Verlag, 1998.
[6] L., Arnold and N. D., Cong. On the simplicity of the Lyapunov spectrum of products of random matrices. Ergod. Theory Dynam. Sys., 17:1005–1025, 1997.
[7] L., Arnold and V., Wishtutz (Eds.). Lyapunov Exponents (Bremen, 1984), volume 1186 of Lecture Notes in Math. Springer, 1986.
[8] A., Avila. Density of positive Lyapunov exponents for quasiperiodic SL(2, ℝ)- cocycles in arbitrary dimension. J. Mod. Dyn., 3:631–636, 2009.
[9] A., Avila and J., Bochi. Lyapunov exponents: parts I & II. Notes of mini-course given at the School on Dynamical Systems, ICTP, 2008.
[10] A., Avila and J., Bochi. Proof of the subadditive ergodic theorem. Preprint www.mat.puc-rio.br/~jairo/.
[11] A., Avila and J., Bochi. A formula with some applications to the theory of Lyapunov exponents. Israel J. Math., 131:125–137, 2002.
[12] A., Avila, A., Eskin and M., Viana. Continuity of Lyapunov exponents of random matrix products. In preparation.
[13] A., Avila, J., Santamaria, and M., Viana. Holonomy invariance: rough regularity and Lyapunov exponents. Astérisque, 358:13–74, 2013.
[14] A., Avila and M., Viana. Simplicity of Lyapunov spectra: a sufficient criterion. Port. Math., 64:311–376, 2007.
[15] A., Avila and M., Viana. Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture. Acta Math., 198:1–56, 2007.
[16] A., Avila and M., Viana. Extremal Lyapunov exponents: an invariance principle and applications. Inventiones Math., 181:115–178, 2010.
[17] A., Avila, M., Viana, and A., Wilkinson. Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows. Preprint www.preprint.impa.br 2011.
[18] L., Barreira and Ya., Pesin. Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Cambridge University Press, 2007.
[19] L., Barreira, Ya., Pesin and J., Schmeling. Dimension and product structure of hyperbolic measures. Ann. of Math., 149:755–783, 1999.
[20] M., Bessa. Dynamics of generic 2-dimensional linear differential systems. J. Differential Equations, 228:685–706, 2006.
[21] M., Bessa. The Lyapunov exponents of generic zero divergence three-dimensional vector fields. Ergodic Theory Dynam. Systems, 27:1445–1472, 2007.
[22] M., Bessa. Dynamics of generic multidimensional linear differential systems. Adv. Nonlinear Stud., 8:191–211, 2008.
[23] M., Bessa and J. L., Dias. Generic dynamics of 4-dimensional C2 Hamiltonian systems. Comm. Math. Phys., 281:597–619, 2008.
[24] M., Bessa and J., Rocha. Contributions to the geometric and ergodic theory of conservative flows. Ergodic Theory Dynam. Sys., 33:1709–1731, 2013.
[25] M., Bessa and H., Vilarinho. Fine properties of lp-cocycles. J. Differential Equations, 256:2337–2367, 2014.
[26] G., Birkhoff. Lattice Theory, volume 25. A.M.S. Colloq. Publ., 1967.
[27] J., Bochi. Discontinuity of the Lyapunov exponents for non-hyperbolic cocycles. Preprint www.mat.puc-rio.br/~jairo/.
[28] J., Bochi. The multiplicative ergodic theorem of Oseledets. Preprint www.mat.puc-rio.br/~jairo/.
[29] J., Bochi. Proof of the Oseledets theorem in dimension 2 via hyperbolic geometry. Preprint www.mat.puc-rio.br/~jairo/.
[30] J., Bochi. Genericity of zero Lyapunov exponents. Ergod. Theory Dynam. Sys., 22:1667–1696, 2002.
[31] J., Bochi. C1-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents. J. Inst. Math. Jussieu, 8:49–93, 2009.
[32] J., Bochi and M., Viana. Pisa lectures on Lyapunov exponents. In Dynamical Systems. Part II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, pages 23–47. Scuola Norm. Sup., 2003.
[33] J., Bochi and M., Viana. Lyapunov exponents: how frequently are dynamical systems hyperbolic? In Modern Dynamical Systems and Applications, pages 271–297. Cambridge University Press, 2004.
[34] J., Bochi and M., Viana. The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. of Math., 161:1423–1485, 2005.
[35] C., Bocker and M., Viana. Continuity of Lyapunov exponents for 2D random matrices. Preprint www.impa.br/~viana/out/bernoulli.pdf.
[36] C., Bonatti, L. J., Diaz, and M., Viana. Dynamics Beyond Uniform Hyperbolicity, volume 102 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, 2005.
[37] C., Bonatti, X., Gomez-Mont, and M., Viana. Généricité d'exposants de Lyapunov non-nuls pour des produits deterministes de matrices. Ann. Inst. H. Poincare Anal. Non Lineaire, 20:579–624, 2003.
[38] C., Bonatti and M., Viana. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math., 115:157–193, 2000.
[39] N., Bourbaki. Algebra. I. Chapters 1-3. Elements of Mathematics (Berlin). Springer-Verlag, 1989. Translated from the French, Reprint of the 1974 edition.
[40] J., Bourgain. Positivity and continuity of the Lyapounov exponent for shifts on Td with arbitrary frequency vector and real analytic potential. J. Anal. Math., 96:313–355, 2005.
[41] J., Bourgain and S., Jitomirskaya. Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J. Statist. Phys., 108:1203–1218, 2002.
[42] M., Cambrainha. Generic symplectic cocycles are hyperbolic. Ph D thesis, IMPA, 2013.
[43] C., Castaing and M., Valadier. Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, 1977.
[44] J. E., Cohen, H., Kesten, and C. M., Newman (Eds.). Random Matrices and their Applications (Brunswick, Maine, 1984), volume 50 of Contemp. Math. Amer. Math. Soc., 1986.
[45] J., Conway. Functions of One Complex Variable. II, volume 159 of Graduate Texts in Mathematics. Springer-Verlag, 1995.
[46] A., Crisanti, G., Paladin, and A., Vulpiani. Products of Random Matrices in Statistical Physics, volume 104 of Springer Series in Solid-State Sciences. Springer-Verlag, 1993. With a foreword by Giorgio Parisi.
[47] D., Damanik. Schrödinger operators with dynamically defined potentials. In preparation.
[48] David, Damanik. Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: a survey of Kotani theory and its applications. In Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon's 60th birthday, volume 76 of Proc. Sympos. Pure Math., pages 539–563. Amer. Math. Soc., 2007.
[49] L. J., Diaz and R., Ures. Persistent homoclinic tangencies at the unfolding of cycles. Ann. Inst. H. Poincare, Anal. Non-lineaire, 11:643–659, 1996.
[50] A., Douady and J. C., Earle. Conformally natural extension of homeomorphisms of the circle. Acta Math., 157:23–48, 1986.
[51] R., Dudley, H., Kunita, F., Ledrappier, and P., Hennequin (Eds.). Ecole d'etede probabilites de Saint-Flour, XII—1982, volume 1097 of Lecture Notes in Math.Springer, 1984.
[52] J., Franks. Anosov diffeomorphisms. In Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), pages 61–93. Amer. Math. Soc., 1970.
[53] N., Friedman. Introduction to Ergodic Theory. Van Nostrand, 1969.
[54] H., Furstenberg. Non-commuting random products. Trans. Amer. Math. Soc., 108:377–428, 1963.
[55] H., Furstenberg. Boundary theory and stochastic processes on homogeneous spaces. In Harmonic Analysis in Homogeneous Spaces, volume XXVI of Proc. Sympos. Pure Math. (Williamstown MA, 1972), pages 193–229. Amer. Math. Soc., 1973.
[56] H., Furstenberg and H., Kesten. Products of random matrices. Ann. Math. Statist., 31:457–469, 1960.
[57] H., Furstenberg and Yu., Kifer. Random matrix products and measures in projective spaces. Israel J. Math, 10:12–32, 1983.
[58] I. Ya., Gol'dsheid and G. A., Margulis. Lyapunov indices of a product of random matrices. UspekhiMat. Nauk., 44:13–60, 1989.
[59] Y., Guivarc'h. Marches aléatories a pas markovien. Comptes Rendus Acad. Sci. Paris, 289:211–213, 1979.
[60] Y., Guivarc'h and A., Raugi. Products of random matrices: convergence theorems. Contemp. Math., 50:31–54, 1986.
[61] H., Hennion. Loi des grands nombres et perturbations pour des produits réductibles de matrices aleatoires indépendantes. Z. Wahrsch. Verw. Gebiete, 67:265–278, 1984.
[62] M., Herman. Une meithode nouvelle pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d'un theoreme d'Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helvetici, 58:453–502, 1983.
[63] A., Herrera. Simplicity of the Lyapunov spectrum for multidimensional continued fraction algorithms. PhD thesis, IMPA, 2009.
[64] F., Rodriguez Hertz, M. A., Rodriguez Hertz, A., Tahzibi, and R., Ures. Maximizing measures for partially hyperbolic systems with compact center leaves. Ergodic Theory Dynam. Sys., 32:825–839, 2012.
[65] E., Hopf. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91:261–304, 1939.
[66] S., Jitomirskaya and C., Marx. Dynamics and spectral theory of quasi-periodic Schrodinger type operators. In preparation.
[67] R., Johnson. Lyapounov numbers for the almost periodic Schrödinger equation. Illinois J. Math., 28:397–419, 1984.
[68] A., Katok. Lyapunov exponents, entropy and periodic points of diffeomorphisms. Publ. Math. IHES, 51:137–173, 1980.
[69] Y., Katznelson and B., Weiss. A simple proof of some ergodic theorems. Israel J. Math., 42:291–296, 1982.
[70] Yu., Kifer. Perturbations of random matrix products. Z. Wahrsch. Verw. Gebiete, 61:83–95, 1982.
[71] Yu., Kifer. Ergodic Theory of Random Perturbations. Birkhauser, 1986.
[72] Yu., Kifer. General random perturbations of hyperbolic and expanding transformations. J. Analyse Math., 47:11–150, 1986.
[73] Yu., Kifer and E., Slud. Perturbations of random matrix products in a reducible case. Ergodic Theory Dynam. Sys., 2:367–382 (1983), 1982.
[74] J., Kingman. The ergodic theorem of subadditive stochastic processes. J. Royal Statist. Soc., 30:499–510, 1968.
[75] O., Knill. The upper Lyapunov exponent of SL(2, R) cocycles: discontinuity and the problem of positivity. In Lyapunov Exponents (Oberwolfach, 1990), volume 1486 of Lecture Notes in Math., pages 86–97. Springer-Verlag, 1991.
[76] O., Knill. Positive Lyapunov exponents for a dense set of bounded measurable SL(2, R)-cocycles. Ergod. Theory Dynam. Sys., 12:319–331, 1992.
[77] S., Kotani. Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators. In Stochastic Analysis, pages 225–248. North Holland, 1984.
[78] H., Kunz and Bernard B., Souillard. Sur le spectre des opeirateurs aux diffeirences finies aleatoires. Comm. Math. Phys., 78:201–246, 1980/1981.
[79] F., Ledrappier. Proprietes ergodiques des mesures de SinaiPubl. Math. I.H.E.S., 59:163–188, 1984.
[80] F., Ledrappier. Quelques proprietes des exposants caracteristiques. volume 1097 of Lect. Notes in Math., pages 305–396, Springer-Verlag, 1984.
[81] F., Ledrappier. Positivity of the exponent for stationary sequences of matrices. In Lyapunov Exponents (Bremen, 1984), volume 1186 of Lect. Notes in Math., pages 56–73. Springer-Verlag, 1986.
[82] F., Ledrappier and G., Royer. Croissance exponentielle de certain produits aleiatorires de matrices. Comptes Rendus Acad. Sci. Paris, 290, 513–514, 1980.
[83] F., Ledrappier and L.-S., YoungThe metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula. Ann. ofMath., 122:509–539, 1985.
[84] F., Ledrappier and L.-S., YoungThe metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. ofMath., 122:540–574, 1985.
[85] A. M., Lyapunov. The General Problem of the Stability of Motion. Taylor & Francis Ltd., 1992. Translated from Edouard Davaux's French translation (1907) of the 1892 Russian original and edited by A.T. Fuller, with an introduction and preface by Fuller, a biography of Lyapunov by V.I. Smirnov, and a bibliography of Lyapunov's works compiled by J.F. Barrett, Lyapunov centenary issue, Reprint of Internat. J. Control 55 (1992), no. 3 [MR1154209 (93e:01035)], With a foreword by Ian Stewart.
[86] R., Mane. A proof of Pesin's formula. Ergod. Theory Dynam. Sys., 1:95–101, 1981.
[87] R., Mane. Lyapunov exponents and stable manifolds for compact transformations. In Geometric Dynamics, volume 1007 of Lect. Notes in Math., pages 522–577. Springer-Verlag, 1982.
[88] R., Mane. Oseledec's theorem from the generic viewpoint. In Procs. International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 1269–1276, Warsaw, 1984. PWN Publ.
[89] R., Mane. Ergodic Theory and Differentiable Dynamics. Springer-Verlag, 1987.
[90] A., Manning. There are no new Anosov diffeomorphisms on tori. Amer.J.Math., 96:422–429, 1974.
[91] S., Newhouse. On codimension one Anosov diffeomorphisms. Amer. J. Math., 92:761–770, 1970.
[92] V. I., Oseledets. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 19:197–231, 1968.
[93] J. C., Oxtoby and S. M., Ulam. Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math., 42:874–920, 1941.
[94] E. Le, Page. Theoremes limites pour les produits de matrices aleatoires. In Probability Measures on Groups (Oberwolfach, 1981), volume 928 of Lecture Notes in Math., pages 258–303. Springer-Verlag, 1982.
[95] É. Le, Page. Reigularitei du plus grand exposant caracteiristique des produits de matrices aleatoires independantes et applications. Ann. Inst. H. Poincare Probab. Statist., 25:109–142, 1989.
[96] J., Palis and S., Smale. Structural stability theorems. In Global Analysis, volume XIV of Proc. Sympos. Pure Math. (Berkeley 1968), pages 223–232. Amer. Math. Soc., 1970.
[97] L., Pastur. Spectral properties of disordered systems in the one-body approximation. Comm. Math. Phys., 75:179–196, 1980.
[98] Y., Peres. Analytic dependence of Lyapunov exponents on transition probabilities. In Lyapunov Exponents (Oberwolfach, 1990), volume 1486 of Lecture Notes in Math., pages 64–80. Springer-Verlag, 1991.
[99] Ya. B., Pesin. Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. USSR. Izv., 10:1261–1302, 1976.
[100] Ya. B., Pesin. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys, 324:55–114, 1977.
[101] M. S., Raghunathan. A proof of Oseledec's multiplicative ergodic theorem. Israel J. Math., 32:356–362, 1979.
[102] V. A., Rokhlin. On the fundamental ideas of measure theory. A. M. S. Transl., 10:1–54, 1962. Transl. from Mat. Sbornik 25 (1949), 107-150. First published by the A. M. S. in 1952 as Translation Number 71.
[103] G., Royer. Croissance exponentielle de produits markoviens de matrices. Ann. Inst. H. Poincare, 16:49–62, 1980.
[104] D., Ruelle. Analyticity properties of the characteristic exponents of random matrix products. Adv. in Math., 32:68–80, 1979.
[105] D., Ruelle. Ergodic theory of differentiable dynamical systems. Inst. Hautes Etudes Sci. Publ. Math., 50:27–58, 1979.
[106] D., Ruelle. Characteristic exponents and invariant manifolds in Hilbert space. Annals of Math., 115:243–290, 1982.
[107] M., Shub. Global Stability of Dynamical Systems. Springer-Verlag, 1987.
[108] B., Simon and M., Taylor. Harmonic analysis on SL(2, R) and smoothness of the density of states in the one-dimensional Anderson model. Comm. Math. Phys., 101:1–19, 1985.
[109] S., Smale. Differentiable dynamical systems. Bull. Am. Math. Soc., 73:747–817, 1967.
[110] A., Tahzibi. Stably ergodic diffeomorphisms which are not partially hyperbolic. Israel J. Math., 142:315–344, 2004.
[111] Ph., Thieullen. Ergodic reduction of random products of two-by-two matrices. J. Anal. Math., 73:19–64, 1997.
[112] M., Viana. Lyapunov exponents and strange attractors. In J.-P., Francoise, G. L., Naber, and S. T., Tsou, editors, Encyclopedia of Mathematical Physics. Elsevier, 2006.
[113] M., Viana. Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. Ann. of Math., 167:643–680, 2008.
[114] M., Viana and K., Oliveira. Fundamentos da Teoria Ergodica. Colecao Fronteiras da Matemaitica. Sociedade Brasileira de Matemaitica, 2014.
[115] M., Viana and J., Yang. Physical measures and absolute continuity for one-dimensional center directions. Ann. Inst. H. Poincare Anal. Non Lineaire, 30:845–877, 2013.
[116] A., Virtser. On products of random matrices and operators. Th. Prob. Appl., 34:367–377, 1979.
[117] P., Walters. A dynamical proof of the multiplicative ergodic theorem. Trans. Amer. Math. Soc., 335:245–257, 1993.
[118] L.-S., Young. Some open sets of nonuniformly hyperbolic cocycles. Ergod. Theory Dynam. Sys., 13(2):409–415, 1993.
[119] L.-S., Young. Ergodic theory of differentiable dynamical systems. In Real and Complex Dynamical Systems, volume NATO ASI Series, C-464, pages 293–336. Kluwer Acad. Publ., 1995.
[120] A. C., Zaanen. Integration. North-Holland Publishing Co., 1967. Completely revised edition of An Introduction to the Theory ofIntegration.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.