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Explore the spectrum rigidity and joint integrability of Anosov systems on tori in this lecture from the Simons Semester on Dynamics. Delve into the strong rigidity properties arising from joint integrability in Anosov diffeomorphisms on tori, focusing on irreducible Anosov diffeomorphisms with split stable bundles. Examine how joint integrability of strong stable and full unstable subbundles leads to fine dominated splitting along weak stable subbundles and Lyapunov exponents rigidity. Investigate the equivalence between geometric rigidity and dynamical spectral rigidity for Anosov diffeomorphisms on tori. Learn about local perturbations of generic irreducible linear automorphisms on T^d and discover how joint integrability implies smooth conjugacy along weak stable bundles.
