Anosov representations are the higher-rank generalization of convex cocompact groups in hyperbolic geometry. Where classical hyperbolic dynamics has one expanding and one contracting direction, Anosov representations in a reductive Lie group G have a richer expanding/contracting structure indexed by a subset of simple roots.
The paper on the spectrum of Anosov representations (arXiv: 2603.24519) constructs a resonance spectrum — a complex analytic variety of codimension 1 in the complexified dual of a split component — that encodes the dynamical information of the representation.
This spectrum governs the meromorphic continuation of zeta functions and Poincaré series. The Ruelle–Pollicott resonances of classical dynamics (poles that control the rate of mixing) become a higher-dimensional geometric object. Under Diophantine conditions on the representation, sharp mixing estimates for the refraction flow are established.
The framework applies to any free abelian cocycle over a hyperbolic flow, positioning it as foundational for higher-rank quantum-classical correspondence: the resonance spectrum is the bridge between the dynamics (classical) and the spectral theory (quantum).
The through-claim: the spectrum of a representation is a geometric object, not a list. In rank one, resonances are isolated points. In higher rank, they form a variety — a codimension-1 submanifold of a complex space. The dynamics doesn't have poles; it has a hypersurface. The dimensionality of the resonance locus matches the rank of the group.
2603.24519. Dynamical systems / Anosov representations / resonance spectrum / Ruelle–Pollicott theory / higher rank.