These intersections form a sort of trellis, web, or infinitely tight mesh... One is struck by the complexity of this figure, which I shall not even attempt to draw.

In this lecture we explore some applications of the Inclination Lemma to chaotic dynamics.

Let $f$ be a dynamical system on a compact manifold $M$, and let $x$ be a hyperbolic periodic point then a point $z \ne x \in W^s(x,f) \cap W^u(x,f)$ is called a transverse homoclinic point for $x$ if the intersection is transverse at $z$:
$$T_zW^s(x,f) + T_z W^u(x,f) = T_z M.$$

The existence of a transverse homoclinic point has profound consequences for the nearby dynamics. The unstable and stable manifolds are forced to seesaw back on themselves infinitely often, creating a “mesh” called a homoclinic tangle.

It was Poincaré who first realised this, during his work on the Three Body Problem in the late 19th century. His investigations led him to imagine a figure of quite breathtaking complexity. The famous quote above is from Volume 3 of Poincaré's series of papers Les Méthodes nouvelles de la mécanique célesté[1]. It sums up his bewilderment at the implications of this discovery. This was arguably humanity's first taste of (mathematical!) chaos, and the development of the entire modern theory of chaotic and hyperbolic dynamics can be traced back to Poincaré's observations.

Today we show how transverse homoclinic points leads to chaotic dynamics and positive entropy. We do this by first looking at the horseshoe map, which is an idealised description of the system near the homoclinic tangle.

1. The original text is available here. ↩︎