In this lecture we explore another surprising consequence of hyperbolicity, known as the Hartman-Grobman Theorem. Namely, if $u$ is a hyperbolic fixed point of a diffeomorphism $f$, then close to $u$ the dynamics of $f$ are conjugate to those of its linearisation $Df(u)$.

In other words, near a hyperbolic fixed point, the dynamics of such a map are entirely governed by those of its linear part. Thus when faced with such a system, one can simplify the picture by linearising it, without losing qualitative dynamical information.

It is important to understand that the Hartman-Grobman Theorem is a purely local statement. It is the only result in the course that does not have a natural global generalisation (i.e. to hyperbolic sets of diffeomorphisms on manifolds).