A backward $\lambda$-Lemma for the forward heat flow

Abstract

The inclination or $\lambda$-Lemma is a fundamental tool in finite dimensional hyperbolic dynamics. In contrast to finite dimension, we consider the forward semi-flow on the loop space of a closed Riemannian manifold $M$ provided by the heat flow. The main result is a backward $\lambda$-Lemma for the heat flow near a hyperbolic fixed point $x$. There are the following novelties. Firstly, infinite versus finite dimension. Secondly, semi-flow versus flow. Thirdly, suitable adaption provides a new proof in the finite dimensional case. Fourthly and a priori most surprisingly, our $\lambda$-Lemma moves the given disk transversal to the unstable manifold backward in time, although there is no backward flow. As a first application we propose a new method to calculate the Conley homotopy index of $x$.

31 pages, 6 figures. Comments most welcome. v2: Theorem 1.2 and Lemma 2.1 slightly improved, corrected typos. v3: minor modifications. To appear in {\it Math. Ann.}

References in corpus (1)

• Morse homology for the heat flow

Cited by in corpus (2)

• Contraction method and Lambda-Lemma
• The backward λ-Lemma and Morse filtrations