By Jaap Eldering
This monograph treats generally hyperbolic invariant manifolds, with a spotlight on noncompactness. those items generalize hyperbolic mounted issues and are ubiquitous in dynamical systems.
First, usually hyperbolic invariant manifolds and their relation to hyperbolic fastened issues and heart manifolds, in addition to, overviews of background and techniques of proofs are provided. moreover, matters (such as uniformity and bounded geometry) coming up because of noncompactness are mentioned in nice element with examples.
The major new end result proven is an explanation of patience for noncompact mostly hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. This extends famous effects by means of Fenichel and Hirsch, Pugh and Shub, and is complementary to noncompactness leads to Banach areas through Bates, Lu and Zeng. alongside the best way, a few new ends up in bounded geometry are acquired and a framework is constructed to investigate ODEs in a differential geometric context.
Finally, the most result's prolonged to time and parameter established structures and overflowing invariant manifolds.
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Additional resources for Normally Hyperbolic Invariant Manifolds: The Noncompact Case
3. We should point out that M is not required to be an embedded submanifold; immersions are allowed as well, see Sect. 2. 21. 1 (Persistence of noncompact NHIMs in bounded geometry) Let k ≥ 2, α ∈ [0, 1] and r = k + α. Let (Q, g) be a smooth Riemannian manifold of k,α k,α a vector field on Q. e. rank(E + ) = 0. Then for each sufficiently small η > 0 there exists a δ > 0 such that for any k,α with v˜ − v 1 < δ, there is a unique submanifold M˜ in the vector field v˜ ∈ Cb,u η-neighborhood of M, such that M˜ is diffeomorphic to M and invariant under the k,α and the distance between M˜ and M can be flow defined by v.
6) are, ignoring arguments, Dk g˜ · D1 + + D2 + Dg k + · · · = D2 − · Dk g + · · · k < 1. The limit on k precisely This leads to a contraction when D2 − · D1 −1 + corresponds to the spectral gap condition, at least when we replace by a sufficiently high iterate N of itself, or in the continuous case, if we take the flow map t at a sufficiently large time t. For the Perron method, the essential form of the derivatives of T is Dk T (x) δx1 , . . , δxk (t) = D t−τ (0) · Dk f (x(τ )) δx1 (τ ), . .
This is similar, but developed independently from Sakamoto’s work [Sak90] in which he used the same ideas to study singular perturbation problems. We improve these results in a couple of ways. First of all, we simplify the basics of the proof by reducing the 24 1 Introduction two-step contraction argument to a single contraction mapping, still written as a composition of two separate maps acting on horizontal curves in M and vertical curves in the normal bundle fiber, respectively. More importantly, we remove the restriction of a trivial product structure X × Y .
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