Invariant manifolds of partially normally hyperbolic invariant manifolds in Banach spaces

We investigate the existence and regularity of locally invariant manifolds near an approximately invariant set that satisfies a geometric hyperbolicity condition with respect to an abstract generalized" dynamical system in Banach spaces. This hyperbolicity framework, which we term partial normal hyperbolicity, bridges the gap between normal hyperbolicity and partial hyperbolicity--concepts previously studied in finite dimensions and specific PDE contexts. Our generalized dynamical system accommodates non-smooth, non-Lipschitz, and even non-mapping" dynamics, making it applicable to both well-posed and ill-posed differential equations. As an illustrative application, we employ our results to analyze the dynamics of whiskered tori.