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A backward $λ$-Lemma for the forward heat flow

The inclination or $λ$-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 $λ$-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 $λ$-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 $x$.