Almost reducibility for families of sequences of matrices

Barreira, Luis; Valls, Claudia

doi:10.1090/proc/15632

Almost reducibility for families of sequences of matrices
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by Luis Barreira and Claudia Valls

Proc. Amer. Math. Soc. 149 (2021), 5223-5236

DOI: https://doi.org/10.1090/proc/15632

Published electronically: September 21, 2021

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Abstract:

We consider the almost reducibility property of a nonautonomous dynamics with discrete time defined by a sequence of matrices. This corresponds to the reduction of the original nonautonomous dynamics to an autonomous dynamics via a coordinate change that preserves the Lyapunov exponents. In particular, we give a characterization of the almost reducibility of a sequence to a diagonal matrix and we use this result to characterize the class of matrices to which a given sequence is almost reducible. We also consider continuous $1$-parameter families of sequences of matrices and we show that the almost reducibility set of such a family is always an $F_{\sigma \delta }$-set. In addition, we show that for any $F_{\sigma \delta }$-set containing zero there exists a family with this set as its almost reducibility set.

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