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Existence and regularity of invariant graphs for cocycles in bundles: partial hyperbolicity case

Volume 555 / 2020

Deliang Chen Dissertationes Mathematicae 555 (2020), 1-176 DOI: 10.4064/dm799-4-2020 Published online: 22 October 2020


We study the existence and regularity of invariant graphs for bundle maps (or bundle correspondences with generating bundle maps motivated by ill-posed differential equations) having some relative partial hyperbolicity on non-trivial and non-locally-compact bundles. The regularity includes (uniform) $ C^0 $ continuity, Hölder continuity and smoothness. To illustrate the power of our results and methods, a number of applications to both well-posed and ill-posed semilinear differential equations and abstract infinite-dimensional dynamical systems are given. These applications include the existence and regularity of different types of invariant foliations (laminations), including strong stable laminations and fake invariant foliations, the existence and regularity of holonomies for cocycles, the $ C^{k,\alpha} $ theorem and decoupling theorem, etc., in a general setting.


  • Deliang Chenhanghai Jiao Tong University
    Shanghai 200240, People’s Republic of China
    College of Mathematics and Physics
    Wenzhou University
    Wenzhou 325035, People’s Republic of China

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