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Lyapunov regularity via singular values


Authors: Luis Barreira and Claudia Valls
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37D99
DOI: https://doi.org/10.1090/tran/6910
Published electronically: May 30, 2017
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Abstract: For a nonautonomous linear dynamics, we study the relation between Lyapunov regularity and the exponential growth rates of the singular values. In particular, for a tempered dynamics, we obtain lower and upper estimates for the Lyapunov exponents in terms of the growth rates. The proof is based on the somewhat unexpected existence of a structure of Oseledets type for any nonregular dynamics. Moreover, we show that any possible values of the Lyapunov exponent and of the growth rates are attained by some bounded sequence of matrices. As an application of our results, we give a simple proof of various characterizations of Lyapunov regularity as well as a new characterization. We consider both discrete and continuous time.


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Additional Information

Luis Barreira
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
Email: barreira@math.tecnico.ulisboa.pt

Claudia Valls
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
Email: cvalls@math.tecnico.ulisboa.pt

DOI: https://doi.org/10.1090/tran/6910
Keywords: Nonautonomous dynamics, regularity, singular values, volume growth
Received by editor(s): March 9, 2015
Received by editor(s) in revised form: January 18, 2016
Published electronically: May 30, 2017
Additional Notes: The authors were supported by FCT/Portugal through UID/MAT/04459/2013
Article copyright: © Copyright 2017 American Mathematical Society