# 48. The Spectral Decomposition Theorem

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.

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