Discontinuous Generalized Hyperbolicity and Structural Stability
Abstract
We study diffeomorphisms of Banach spaces satisfying generalized hyperbolicity: the stable and unstable subspaces are invariant only up to inclusion, need not depend continuously on the point, and expansivity may fail. On closed manifolds, this condition is equivalent to Axiom~A with strong transversality. We introduce a continuous-selection method that converts uniform orbitwise estimates into bounded continuous solutions of cohomological equations for generalized-hyperbolic cocycles. For diffeomorphisms in the uniform global C1 category, this yields continuous semiconjugacies in both directions under sufficiently small bounded Lipschitz perturbations, and topological stability in the sense of Walters. For the infinite product of one-dimensional Morse-Smale systems on ℓp(ℤ), 1≤ p≤∞, we construct compatible forward and reverse cohomological solution operators and prove structural stability, including for perturbations that couple coordinates, although the product is not expansive. For p=∞ this gives a structurally stable diffeomorphism with uncountably many hyperbolic fixed points, in contrast with the finitely many basic sets of Axiom~A diffeomorphisms. The cocycle analysis yields Lipschitz shadowing for the induced operator on bounded continuous vector fields.
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BibTeX
@misc{tikhomirov2026discontinuous,
title = {{Discontinuous Generalized Hyperbolicity and Structural Stability}},
author = {Sergey Tikhomirov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04179},
eprint = {2610.04179},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04179},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Discontinuous Generalized Hyperbolicity and Structural Stability
AU - Sergey Tikhomirov
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.04179
UR - https://arxiv.org/abs/2610.04179
AB - We study diffeomorphisms of Banach spaces satisfying generalized hyperbolicity: the stable and unstable subspaces are invariant only up to inclusion, need not depend continuously on the point, and expansivity may fail. On closed manifolds, this condition is equivalent to Axiom~A with strong transversality. We introduce a continuous-selection method that converts uniform orbitwise estimates into bounded continuous solutions of cohomological equations for generalized-hyperbolic cocycles. For diffeomorphisms in the uniform global C^1 category, this yields continuous semiconjugacies in both directions under sufficiently small bounded Lipschitz perturbations, and topological stability in the sense of Walters. For the infinite product of one-dimensional Morse-Smale systems on ℓ^p(ℤ), 1≤ p≤∞, we construct compatible forward and reverse cohomological solution operators and prove structural stability, including for perturbations that couple coordinates, although the product is not expansive. For p=∞ this gives a structurally stable diffeomorphism with uncountably many hyperbolic fixed points, in contrast with the finitely many basic sets of Axiom~A diffeomorphisms. The cocycle analysis yields Lipschitz shadowing for the induced operator on bounded continuous vector fields.
ER -