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Transactions of the American Mathematical Society

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Hilbert space Lyapunov exponent stability


Authors: Gary Froyland, Cecilia González-Tokman and Anthony Quas
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37H15
DOI: https://doi.org/10.1090/tran/7521
Published electronically: May 20, 2019
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Abstract: We study cocycles of compact operators acting on a separable Hilbert space and investigate the stability of the Lyapunov exponents and Oseledets spaces when the operators are subjected to additive Gaussian noise. We show that as the noise is shrunk to 0, the Lyapunov exponents of the perturbed cocycle converge to those of the unperturbed cocycle, and the Oseledets spaces converge in probability to those of the unperturbed cocycle. This is, to our knowledge, the first result of this type with cocycles taking values in operators on infinite-dimensional spaces. The infinite dimensionality gives rise to a number of substantial difficulties that are not present in the finite-dimensional case.


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Additional Information

Gary Froyland
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
Email: g.froyland@unsw.edu.au

Cecilia González-Tokman
Affiliation: School of Mathematics & Physics, The University of Queensland, St. Lucia, Queensland 4072, Australia
Email: cecilia.gt@uq.edu.au

Anthony Quas
Affiliation: Department of Mathematics & Statistics, University of Victoria, Victoria, BC, Canada V8W 2Y2
Email: aquas@uvic.ca

DOI: https://doi.org/10.1090/tran/7521
Received by editor(s): October 17, 2017
Received by editor(s) in revised form: January 29, 2018
Published electronically: May 20, 2019
Additional Notes: The first and third authors acknowledge partial support from the Australian Research Council (DP150100017).
The research of the second author has been supported by the Australian Research Council (DE160100147).
The third author also acknowledges the support of NSERC
Article copyright: © Copyright 2019 American Mathematical Society