Contents/conteúdo

Mathematics Department Técnico Técnico

LisMath Seminar  RSS

26/04/2021, 09:00 — 09:30 — Online
João Rijo, LisMath, Instituto Superior Técnico, Universidade de Lisboa

Nonuniform Hyperbolicity in Difference Equations: Admissibility and Infinite Delay

We consider a nonautonomous dynamical system given by a sequence of bounded linear operators acting on a Banach space. We introduce the notion of an exponential dichotomy which is central in the stability theory of dynamical systems. Our results give a characterization of the existence of an exponential dichotomy in terms of the invertibility of a certain linear operator between so-called admissible spaces. Using this characterization, we show that the notion of an exponential dichotomy is robust for sufficiently small linear perturbations.

We also introduce the notion of an exponential dichotomy for difference equations with infinite delay. This requires considering an appropriate class of phase spaces that are Banach spaces of sequences satisfying a certain axiom motivated by the work of Hale and Kato for continuous time. We present a result that establishes the existence of stable manifolds for any sufficiently small perturbation of a difference equation having an exponential dichotomy.

Finally, we briefly describe the formulation of the previous results for the more general case of a tempered exponential dichotomy. This is a nonuniform version of an exponential dichotomy that is ubiquitous in the context of ergodic theory.

References

[1] L. Barreira, J. Rijo, C. Valls. Characterization of tempered exponential dichotomies. J. Korean Math. Soc., 57(1):171–194, 2020.

[2] L. Barreira, J. Rijo, C. Valls. Stable manifolds for difference equations with infinite delay. J. Difference Equ. Appl., 26(9-10):1266–1287, 2020.


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