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

TL;DR

This paper extends the classical inclination or λ-Lemma to the infinite-dimensional setting of the heat flow on loop spaces, providing new insights and tools for analyzing hyperbolic fixed points in this context.

Contribution

It introduces a backward λ-Lemma for the heat flow on loop spaces, adapting finite-dimensional techniques to infinite dimensions and offering a novel method for computing the Conley index.

Findings

Established a backward λ-Lemma for the heat flow near hyperbolic fixed points.

Provided a new proof of the finite-dimensional λ-Lemma using the infinite-dimensional approach.

Proposed a new method to calculate the Conley homotopy index of hyperbolic fixed points.

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 $x$.