Global periodic-data rigidity for irreducible toral automorphisms

Abstract

We prove a global \(C^{1+\text{Hölder}}\)-rigidity theorem for Anosov diffeomorphisms of tori with irreducible linearization. Let \(f:\mathbb T^N\to\mathbb T^N\) be a \(C^2\) Anosov diffeomorphism with linearization \(A\in GL(N,\mathbb Z)\), and assume that \(A\) is irreducible. If, for every periodic point \(p=f^n p\), the linear maps \(Df_p^n\) and \(A^n\) are conjugate, then the Franks--Manning conjugacy between \(f\) and \(A\) is \(C^{1+\text{Hölder}}\). Thus, in the irreducible case, periodic data completely characterize global \(C^{1+\text{Hölder}}\)-rigidity. The proof does not assume conformality, uniform quasiconformality, simplicity of the spectrum, or any restriction on Lyapunov multiplicities. The main ingredient is a new partial-to-global rigidity mechanism combining geometric and analytic arguments. We first obtain partial cocycle rigidity on canonical conformal layers inside the Lyapunov blocks by geometric methods, and then promote this partial rigidity to full regularity of the conjugacy along the Lyapunov blocks by analytic methods. The same method yields a local rigidity theorem for \(C^1\)-small \(C^{1+\text{Hölder}}\) perturbations of \(A\).