Joint integrability and spectral rigidity for Anosov diffeomorphisms

Abstract Let be an Anosov diffeomorphism whose linearization is irreducible. Assume that is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable subbundle and the unstable subbundle are jointly integrable, then is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization . Moreover, admits the finest dominated splitting in the weak stable subbundle with dimensions matching those for , and it has spectral rigidity along all these subbundles. In dimension 4, we also obtain a similar result by grouping the weak stable and unstable subbundles together as a center subbundle and assuming joint integrability of the strong stable and unstable subbundles. As an application, we show that for every symplectic diffeomorphism that is ‐close to an irreducible nonconformal automorphism , the extremal subbundles of are jointly integrable if and only if is smoothly conjugate to .