In this lecture we investigate dynamical systems $f \colon M \to M$ with the property that $\overline{\mathsf{per}(f)}$ is hyperbolic. We prove the Spectral Decomposition Theorem[1] of Smale, and use this to show that if $\overline{\mathsf{per}(f)}$ is hyperbolic then it is automatically isolated.

The proofs in this lecture are all super fun[2]. We repeatedly use our two “favourite” tactics for attacking problems in hyperbolic dynamics: the Anosov Closing Lemma from Lecture 45, and the Inclination Lemma from Lecture 46. This is highly satisfying.

1. Despite the name, this theorem does not refer to the spectra of anything. ↩︎

2. For me. ↩︎