Generalized $(C,\lambda )$-structure for nonlinear diffeomorphisms of Banach spaces

We introduce the notion of a generalized $(C,\lambda )$-structure, which generalizes hyperbolicity to nonlinear dynamics in Banach spaces. The main novelties are that we allow the hyperbolic splitting to be discontinuous, and that in the invariance condition we assume only inclusions rather than equalities for both the stable and unstable subspaces. This allows us to cover Morse-Smale systems and generalized hyperbolicity. We suggest that generalized $(C,\lambda )$-structure for infinite-dimensional dynamics plays a role analogous to that of ``Axiom A and the strong transversality condition’’ for dynamics on compact manifolds. For diffeomorphisms of a reflexive Banach space, we show that generalized $(C,\lambda )$-structure implies Lipschitz (periodic) shadowing and is robust under $C^1$-small perturbations. Assuming that generalized $(C,\lambda )$-structure is continuous for diffeomorphisms on an arbitrary Banach space we obtain a weak form of structural stability: the diffeomorphism is semi-conjugate in both directions with any of its $C^1$-small perturbation.