Journal of Modern Dynamics (JMD)

A concise proof of the multiplicative ergodic theorem on Banach spaces

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

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Cecilia González-Tokman - School of Mathematics and Physics, The University of Queensland, St Lucia QLD 4072, Australia (email)
Anthony Quas - Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., V8W 3R4, Canada (email)

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.

Received: July 2014;      Revised: May 2015;      Available Online: September 2015.