Contraction method and Lambda-Lemma

TL;DR

This paper offers a new proof of the lambda-lemma for finite-dimensional gradient flows using the contraction method, providing a quantitative description of stable and unstable foliations and their dynamical properties.

Contribution

It generalizes the contraction method proof of the lambda-lemma, enabling the construction of stable and unstable foliations with dynamical thickenings in finite-dimensional gradient flows.

Findings

Quantitative description of (un)stable foliations

Construction of dynamical thickenings for leaves

Extension of contraction method to lambda-lemma

Abstract

We reprove the $\lambda$-Lemma for finite dimensional gradient flows by generalizing the well-known contraction method proof of the local (un)stable manifold theorem. This only relies on the forward Cauchy problem. We obtain a rather quantitative description of (un)stable foliations which allows to equip each leaf with a copy of the flow on the central leaf -- the local (un)stable manifold. These dynamical thickenings are key tools in our recent work [Web]. The present paper provides their construction.