Some results on linear dynamical systems

Date

2023-09-07

Authors

Lee, George

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Abstract

A linear cocycle is an object that arises naturally in the study of dynamical systems and statistics. Oseledets’ multiplicative ergodic theorem [22] guarantees a decompo- sition of a linear space of states into equivariant subspaces that grow logarithmically at rates corresponding to the Lyapunov exponents. Theorem 66 is the main result of this thesis, a semi-invertible version of this theorem: the ergodic system is invertible, the state space is a separable Banach space and the cocycle is strongly measurable and forward integrable, but with no invertability or injectivity assumptions. Past results have implicitly or explicitly made extra assumptions on the underlying cocycle, but in this work no injectivity or separability of the dual is assumed, and no prior version of the result or other overly unconstructive machinery is used in obtaining the direct sum decomposition of the state spacs

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Keywords

Dynamical systems, Ergodic theory, Mathematics

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