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