I now know what I’m working on for my REU project; I’ll be studying (with two other undergraduates) a type of cohomology for dynamical systems. Misha Guysinsky, our mentor, has not explained the specific problem yet—perhaps that’ll come when we meet with him on Thursday. So I’ve spent the last weekend trying to learn a few basic facts about (especially hyperbolic) dynamical systems, which I will try to explain here.

1. Why do we care about hyperbolicity?

So, first a definition: let ${f: M \rightarrow M}$ be a ${C^1}$-morphism of a smooth manifold ${M}$. Suppose ${p \in M}$ is a fixed point. Then ${p}$ is called hyperbolic if the derivative ${Df_p: T_p(M) \rightarrow T_p(M)}$ has no eigenvalues on the unit circle. This comes from linear algebra: an endomorphism of a vector space is called hyperbolic if its eigenvalues are off the unit circle. Hyperbolicity is an important condition in dynamics, and I want to illustrate this with a few examples. (more…)