Today we state and prove the famous Inclination Lemma (also sometimes called the $\lambda$-Lemma) of Palis.

Roughly speaking, the Inclination Lemma says that successive images of any disc transverse to the stable manifold of a hyperbolic fixed point eventually pile up on the unstable manifold.

If the stable and unstable manifolds of a given hyperbolic fixed point intersect transversely, this “piling up” phenomena can be used to create chaotic dynamics in a neighbourhood of the fixed point. We will investigate this next lecture.