A concise proof of themultiplicative ergodic theorem on Banach spaces

Cecilia  González-Tokman; Anthony Quas

doi:10.3934/jmd.2015.9.237

Article Contents

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

This issue Previous Article Hofer's length spectrum of symplectic surfaces Next Article Iterated identities and iterational depth of groups

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
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: August 2015

Abstract Related Papers Cited by

Abstract

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,
multiplicative ergodic theorem,
infinite dimensional random dynamical systems.

Mathematics Subject Classification: Primary: 37H15; Secondary: 37L55.

Citation:

Related Papers

Cited by

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.