LISMATH
Lisbon Mathematics PhD

The LisMath Seminar

The purpose of the LisMath seminar is to provide an initiation into research as well as to make students train their oral skills.  Each LisMath student will be asked to give a talk, based on research papers, chosen from a list covering a variety of research topics. The seminar should be comprehensible to everyone. The student will be asked to make an effort to explain why he/she finds the topic interesting, and how it fits into the broader research picture. The LisMath seminar will thus help broadening the students training.
 
The LisMath seminar takes place on a weekly basis in the Spring semester. Attendance is mandatory for LisMath students. Venue: Wednesday 17h-18h, alternating between FCUL (seminar room 6.2.33 of the Department of Mathematics) and IST (seminar room P9 of the Department of Mathematics) except for 26/6/2018 and 15/7/2019, LisMath Seminar Day, when all sessions will be held at the former location.

– Europe/Lisbon — 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.