Let $(M,m)$ denote a compact Riemannian manifold, and suppose $f \colon M \to M$ is a dynamical system on $M$. Using the exponential map $\exp \colon TM \to M$ of $m$, we can lift $f$ to a map $ \widehat{f} $ on the tangent bundle $TM$, defined by

$$ \widehat{f}(x,v) := \exp_{f(x)}^{-1} \circ f \circ \exp_x(v)$$

(this expression is well defined for $\| v \|$ small enough).

Recall in the proof of the Hartman-Grobman Theorem we used the fact that, near a hyperbolic fixed point, $ f - Df$ is Lipschitz small. Today we do the same thing, only with $f$ replaced with $\widehat{f}$. And instead of looking for fixed points, we look for invariant vector fields.

Comments and questions?