Hyperbolicity for Dynamics on Banach spaces: Generalized (C, λ)-structure
Sergey Tikhomirov (PUC-Rio)

We introduce a notion of generalized (C, λ)-structure for nonlinear diffeomorphisms of Banach spaces. The main differences to the classical notion of hyperbolicity are that we allow the hyperbolic splitting to be discontinuous and in invariance condition assume only inclusions instead of equalities for both stable and unstable subspace. These aspects allow us to cover Morse- Smale systems and generalized hyperbolicity. We suggest that the generalized (C, λ)-structure for infinite-dimensional dynamics plays a role similar to “Axiom A and strong transversality condition” for dynamics on compact manifolds. For diffeomorphisms of reflexive Banach space we showed that generalized (C, λ)-structure implies Lipschitz (periodic) shadowing and is robust under C1-small perturbations. Assuming that generalized (C, λ)-structure is continuous for diffeomorphisms of arbitrary Banach spaces we obtain a weak form of structural stability: the diffeomorphism is semiconjugated from both sides with any C1-small perturbation.