Poster Session

Global linearization and fiber bundle structure of invariant manifolds

Matthew Kvalheim
University of Michigan
USA
Co-Author(s):    Jaap Eldering, Shai Revzen
Abstract:
We study properties of the global (center-)stable manifold of a compact normally attracting invariant manifold (NAIM) for a flow, the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. As a slight generalization, we allow the NAIM to be inflowing invariant, which means that the boundary of the NAIM might be nonempty, but the vector field points inwards there. We extend a classical result of Pugh and Shub for the boundaryless case, showing that the flow near an inflowing NAIM is "linearizable" or topologically conjugate to its linearization at the NAIM. Moreover, we give conditions ensuring the existence of a $C^k$ conjugacy. We also show that a $C^k$ locally linearizable inflowing NAIM is automatically globally $C^k$ linearizable, with the conjugacy defined on the entire global (center)-stable manifold. We finally show that under weaker $k$-center bunching conditions, the global stable foliation has a $C^k$ disk bundle structure, which can be interpreted as a weak form of our global linearization results. We apply our results to geometric singular perturbation theory by extending the domain of the Fenichel Normal Form to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.