American Institute of Mathematical Sciences

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2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237

A concise proof of the multiplicative ergodic theorem on Banach spaces

Cecilia González-Tokman 1, and  Anthony Quas 2,

1. 

School of Mathematics and Physics, The University of Queensland, St Lucia QLD 4072, Australia
2. 

Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., V8W 3R4

Received  July 2014 Revised  May 2015 Published  September 2015

We give a new proof of a multiplicative ergodic theorem for quasi-compact operators on Banach spaces with a separable dual. Our proof works by constructing the finite-codimensional `slow' subspaces (those where the growth rate is dominated by some $\lambda_i$), in contrast with earlier infinite-dimensional multiplicative ergodic theorems which work by constructing the finite-dimensional fast subspaces. As an important consequence for applications, we are able to get rid of the injectivity requirements that appear in earlier works.
Keywords: Oseledets theorem, infinite dimensional random dynamical systems., multiplicative ergodic theorem.
Mathematics Subject Classification: Primary: 37H15; Secondary: 37L5.
Citation: 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
References:
[1]

A. Blumenthal, A volume-based approach to the multiplicative ergodic theorem on Banach spaces, arXiv:1502.06554.

[2]

T. S. Doan, Lyapunov Exponents for Random Dynamical Systems, PhD thesis, Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2009.

[3]

G. Froyland, S. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756. doi: 10.1017/S0143385709000339.

[4]

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

[5]

T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[6]

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.

[7]

R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer, Berlin, 1983, 522-577. doi: 10.1007/BFb0061433.

[8]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.

[9]

G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics, No. 94, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511662454.

[10]

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

[11]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290. doi: 10.2307/1971392.

[12]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97.

[13]

P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, 25, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511608735.

show all references

References:
[1]

A. Blumenthal, A volume-based approach to the multiplicative ergodic theorem on Banach spaces, arXiv:1502.06554.

[2]

T. S. Doan, Lyapunov Exponents for Random Dynamical Systems, PhD thesis, Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2009.

[3]

G. Froyland, S. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756. doi: 10.1017/S0143385709000339.

[4]

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

[5]

T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[6]

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.

[7]

R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer, Berlin, 1983, 522-577. doi: 10.1007/BFb0061433.

[8]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.

[9]

G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics, No. 94, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511662454.

[10]

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

[11]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290. doi: 10.2307/1971392.

[12]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97.

[13]

P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, 25, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511608735.

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