SEE Rio 

Palestrante: Sergey Tikhomirov (PUC-Rio)

Coordenadas: Quarta-feira, 11 de março, 17h na sala L856

Título: Generalized $(C, \lambda)$-structure in Banach spaces: shadowing, robustness and semi-structural stability.

Resumo: Hyperbolicity is a central concept in the theory of structural stability on manifolds.  
This talk is devoted to the notion of hyperbolicity in Banach spaces.  
Relatively recently [1, 2], the concept of \emph{generalized hyperbolicity} for linear mappings in Banach spaces was introduced, and it was shown in particular that it implies both the shadowing property and an analogue of the Grobman--Hartman theorem.

In this talk, combining the ideas of the $(C, \lambda)$-structure [3] for finite-dimensional manifolds with the notion of generalized hyperbolicity, we introduce a \emph{generalized  $(C, \lambda)$ -structure} for nonlinear diffeomorphisms of Banach spaces [4].  
The key novelty is the possibility of discontinuous dependence of "hyperbolic splitting" on a point and "dimension variability" along trajectory for both stable and unstable foliations.
We prove that the generalized  $(C, \lambda)$ -structure implies:
-- finite Lipschitz shadowing in arbitrary Banach spaces,
-- infinite Lipschitz shadowing and periodic shadowing in reflexive Banach spaces.
-- robustness of generalized  $(C, \lambda)$ -structure under $C^1$ small perturbations.
Situation with structural stability is more involved, we managed to prove only its weak version (semi-conjugacy from both sides with $C^1$-small perturbations) under extra assumption of continuity of the splitting.

[1] N. Bernardes, P. Cirilo, U. Darji, A. Messaoudi, E. Pujals, Expansivity and shadowing in linear dynamics, J. Math. Anal. Appl. 461 (2018) 796–816.
[2] P. Cirilo, B. Gollobit, and Enrique Pujals. Dynamics of generalized hyperbolic linear operators. Advances in Mathematics 387 (2021): 107830.
[3] S. Yu Pilyugin. Generalizations of the notion of hyperbolicity. J. Difference Equ. Appl. 12 (2006), 271–282. [4] S. Tikhomirov. Generalized $(C, \lambda)$-structure for nonlinear diffeomorphisms of Banach spaces. https://arxiv.org/abs/2510.05499