May  2016, 36(5): 2377-2403. doi: 10.3934/dcds.2016.36.2377

A volume-based approach to the multiplicative ergodic theorem on Banach spaces

1. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, United States

Received  July 2014 Revised  September 2015 Published  October 2015

A volume growth-based proof of the Multiplicative Ergodic Theorem for Banach spaces is presented, following the approach of Ruelle for cocycles acting on a Hilbert space. As a consequence, we obtain a volume growth interpretation for the Lyapunov exponents of a Banach space cocycle.
Citation: Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377
References:
[1]

R. R. Akhmerov, M. Kamenskii, A. Potapov, A. Rodkina and B. Sadovskii, Measures of noncompactness and condensing operators, Operator Theory, 55 (1992), 1-244. doi: 10.1007/978-3-0348-5727-7.  Google Scholar

[2]

L. Arnold, Random Dynamical Systems, Springer, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[3]

E. Berkson, Some metrics on the subspaces of a banach space, Pacific J. Math, 13 (1963), 7-22. doi: 10.2140/pjm.1963.13.7.  Google Scholar

[4]

B. Bollobas, Linear Analysis, an Introductory Course, Cambridge University Press, 1999. doi: 10.1017/CBO9781139168472.  Google Scholar

[5]

H. Busemann, Intrinsic area, Annals of Mathematics, 48 (1947), 234-267. doi: 10.2307/1969168.  Google Scholar

[6]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[7]

M. M. Day, Normed Linear Spaces, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973.  Google Scholar

[8]

G. B. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.  Google Scholar

[9]

G. Froyland, S. Lloyd and A. Quas, A semi-invertible oseledets theorem with applications to transfer operator cocycles, Discrete and Continuous Dynamical Systems, 33 (2013), 3835-3860. doi: 10.3934/dcds.2013.33.3835.  Google Scholar

[10]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on banach spaces, Journal of Modern Dynamics, 9 (2015), 237-255. doi: 10.3934/jmd.2015.9.237.  Google Scholar

[11]

C. González-Tokman and A. Quas, A semi-invertible operator oseledets theorem, Ergodic Theory and Dynamical Systems, 34 (2014), 1230-1272. doi: 10.1017/etds.2012.189.  Google Scholar

[12]

E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, American Mathematical Soc., 1957.  Google Scholar

[13]

T. Kato, Perturbation Theory for Linear Operators, Springer, 1995.  Google Scholar

[14]

H. Kober, A theorem on banach spaces, Compositio Mathematica, 7 (1940), 135-140. Google Scholar

[15]

U. Krengel and A. Brunel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. doi: 10.1515/9783110844641.  Google Scholar

[16]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc., 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[17]

R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in Geometric Dynamics, Lecture Notes in Math., 1007, Springer, 1983, 522-577. doi: 10.1007/BFb0061433.  Google Scholar

[18]

V. I. Oseledets, A multiplicative ergodic theorem. characteristic lyapunov exponents of dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva, 19 (1968), 179-210.  Google Scholar

[19]

A. Pietsch, Eigenvalues and S-Numbers, Cambridge University Press Cambridge, 1987.  Google Scholar

[20]

M. S. Raghunathan, A proof of oseledec's multiplicative ergodic theorem, Israel Journal of Mathematics, 32 (1979), 356-362. doi: 10.1007/BF02760464.  Google Scholar

[21]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 50 (1979), 27-58.  Google Scholar

[22]

D. Ruelle, Characteristic exponents and invariant manifolds in hilbert space, Annals of Mathematics, 115 (1982), 243-290. doi: 10.2307/1971392.  Google Scholar

[23]

H. Rund, The Differential Geometry of Finsler Spaces, Springer, 1959.  Google Scholar

[24]

M. Schechter, Principles of Functional Analysis, American Mathematical Soc., 1973.  Google Scholar

[25]

B.-M. D. T. Son, Lyapunov Exponents for Random Dynamical Systems, PhD thesis, Technische Universität Dresden, 2009. Google Scholar

[26]

R. Temam, Infinite Dimensonal Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[27]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts exposants de Lyapounov. Entropie. Dimension, Annales de l'institut Henri Poincaré (C) Analyse Non Linéaire, 4 (1987), 49-97.  Google Scholar

[28]

P. Walters, A dynamical proof of the multiplicative ergodic theorem, Transactions of the American Mathematical Society, 335 (1993), 245-257. doi: 10.1090/S0002-9947-1993-1073779-7.  Google Scholar

[29]

P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, 1991. doi: 10.1017/CBO9780511608735.  Google Scholar

show all references

References:
[1]

R. R. Akhmerov, M. Kamenskii, A. Potapov, A. Rodkina and B. Sadovskii, Measures of noncompactness and condensing operators, Operator Theory, 55 (1992), 1-244. doi: 10.1007/978-3-0348-5727-7.  Google Scholar

[2]

L. Arnold, Random Dynamical Systems, Springer, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[3]

E. Berkson, Some metrics on the subspaces of a banach space, Pacific J. Math, 13 (1963), 7-22. doi: 10.2140/pjm.1963.13.7.  Google Scholar

[4]

B. Bollobas, Linear Analysis, an Introductory Course, Cambridge University Press, 1999. doi: 10.1017/CBO9781139168472.  Google Scholar

[5]

H. Busemann, Intrinsic area, Annals of Mathematics, 48 (1947), 234-267. doi: 10.2307/1969168.  Google Scholar

[6]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[7]

M. M. Day, Normed Linear Spaces, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973.  Google Scholar

[8]

G. B. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.  Google Scholar

[9]

G. Froyland, S. Lloyd and A. Quas, A semi-invertible oseledets theorem with applications to transfer operator cocycles, Discrete and Continuous Dynamical Systems, 33 (2013), 3835-3860. doi: 10.3934/dcds.2013.33.3835.  Google Scholar

[10]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on banach spaces, Journal of Modern Dynamics, 9 (2015), 237-255. doi: 10.3934/jmd.2015.9.237.  Google Scholar

[11]

C. González-Tokman and A. Quas, A semi-invertible operator oseledets theorem, Ergodic Theory and Dynamical Systems, 34 (2014), 1230-1272. doi: 10.1017/etds.2012.189.  Google Scholar

[12]

E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, American Mathematical Soc., 1957.  Google Scholar

[13]

T. Kato, Perturbation Theory for Linear Operators, Springer, 1995.  Google Scholar

[14]

H. Kober, A theorem on banach spaces, Compositio Mathematica, 7 (1940), 135-140. Google Scholar

[15]

U. Krengel and A. Brunel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. doi: 10.1515/9783110844641.  Google Scholar

[16]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc., 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[17]

R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in Geometric Dynamics, Lecture Notes in Math., 1007, Springer, 1983, 522-577. doi: 10.1007/BFb0061433.  Google Scholar

[18]

V. I. Oseledets, A multiplicative ergodic theorem. characteristic lyapunov exponents of dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva, 19 (1968), 179-210.  Google Scholar

[19]

A. Pietsch, Eigenvalues and S-Numbers, Cambridge University Press Cambridge, 1987.  Google Scholar

[20]

M. S. Raghunathan, A proof of oseledec's multiplicative ergodic theorem, Israel Journal of Mathematics, 32 (1979), 356-362. doi: 10.1007/BF02760464.  Google Scholar

[21]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 50 (1979), 27-58.  Google Scholar

[22]

D. Ruelle, Characteristic exponents and invariant manifolds in hilbert space, Annals of Mathematics, 115 (1982), 243-290. doi: 10.2307/1971392.  Google Scholar

[23]

H. Rund, The Differential Geometry of Finsler Spaces, Springer, 1959.  Google Scholar

[24]

M. Schechter, Principles of Functional Analysis, American Mathematical Soc., 1973.  Google Scholar

[25]

B.-M. D. T. Son, Lyapunov Exponents for Random Dynamical Systems, PhD thesis, Technische Universität Dresden, 2009. Google Scholar

[26]

R. Temam, Infinite Dimensonal Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[27]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts exposants de Lyapounov. Entropie. Dimension, Annales de l'institut Henri Poincaré (C) Analyse Non Linéaire, 4 (1987), 49-97.  Google Scholar

[28]

P. Walters, A dynamical proof of the multiplicative ergodic theorem, Transactions of the American Mathematical Society, 335 (1993), 245-257. doi: 10.1090/S0002-9947-1993-1073779-7.  Google Scholar

[29]

P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, 1991. doi: 10.1017/CBO9780511608735.  Google Scholar

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