Discontinuous Generalized Hyperbolicity and Structural Stability

Abstract

We introduce a new method based on Michael's continuous selection theorem for stability of Banach-space dynamics with possibly discontinuous generalized-hyperbolic splittings. The method converts uniform orbitwise solvability estimates into bounded continuous solutions of cohomological equations. For generalized-hyperbolic diffeomorphisms in the uniform global $C^1$ category, it yields continuous semiconjugacies in both directions under sufficiently small bounded Lipschitz perturbations; surjectivity is not asserted. For an infinite product of one-dimensional Morse-Smale systems, we construct compatible forward and reverse cohomological solution operators and obtain full structural stability without expansivity. This result holds on every $l^p(Z)$, $1\leq p\leq\infty$, including perturbations that couple coordinates. These products have no compact global attractor and, for $p=\infty$, have uncountably many hyperbolic fixed points. The continuous-selection argument underlying the general semiconjugacy theorem is developed for generalized-hyperbolic cocycles, yielding bounded continuous solvability of cohomological equations. The cocycle analysis also gives Lipschitz shadowing for the induced operator on bounded continuous vector fields.