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 and 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.

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[29]

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

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.

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[29]

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

[1]

J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731

[2]

Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237

[3]

Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149

[4]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

[5]

Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303

[6]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187

[7]

Chunyan Zhao, Chengkui Zhong, Xiangming Zhu. Existence of compact $ \varphi $-attracting sets and estimate of their attractive velocity for infinite-dimensional dynamical systems. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022051

[8]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91

[9]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[10]

Savin Treanţă. On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces. Evolution Equations and Control Theory, 2022, 11 (3) : 827-836. doi: 10.3934/eect.2021027

[11]

Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations and Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207

[12]

Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control and Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83

[13]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066

[14]

Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807

[15]

Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166

[16]

Paolo Perfetti. A Nekhoroshev theorem for some infinite--dimensional systems. Communications on Pure and Applied Analysis, 2006, 5 (1) : 125-146. doi: 10.3934/cpaa.2006.5.125

[17]

Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012

[18]

Sergey V Lototsky, Henry Schellhorn, Ran Zhao. An infinite-dimensional model of liquidity in financial markets. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 117-138. doi: 10.3934/puqr.2021006

[19]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098

[20]

Birgit Jacob, Hafida Laasri. Well-posedness of infinite-dimensional non-autonomous passive boundary control systems. Evolution Equations and Control Theory, 2021, 10 (2) : 385-409. doi: 10.3934/eect.2020072

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]