Sergey Tikhomirov

Pontificia Universidade Católica do Rio de Janeiro, Brazil

Title: Shadowing in nonlinear dynamics in Banach spaces: generalized $(C, \lambda)$-structure 

Abstract: 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.  

Under appropriate regularity assumptions, we prove that the generalized  $(C, \lambda)$ -structure implies:

 -- finite Lipschitz shadowing in arbitrary Banach spaces,

 -- infinite Lipschitz shadowing in reflexive Banach spaces.

Using those shadowing results we also prove robustness of \emph{generalized  $(C, \lambda)$ -structure} under $C^1$ small perturbations.

[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.