Ruelle-Taylor resonances of Anosov actions

The paper proves that joint Ruelle (Ruelle–Taylor) resonances for smooth abelian Anosov actions form a discrete subset of a*_C, lie in the intersection of half-spaces Re λ(A) ≤ 0 for A in the positive Weyl chamber W, and have finite-dimensional joint resonant spaces (Theorem 1), via a cohomological/Koszul-complex and joint Taylor-spectrum parametrix approach . The candidate solution establishes the same statements by a different route: it fixes one Anosov generator A0, uses the one-parameter Pollicott–Ruelle theory to get finite-dimensional resonant spaces for X_{A0}, invokes commutativity to restrict all X_A to those spaces, and then uses finite-dimensional linear algebra (simultaneous triangularization) to read off joint eigen-characters and hence discreteness/finite multiplicity; it obtains the same half-space bound using the one-parameter characterization of resonances (WF(u) ⊂ E_u^*) for each B ∈ W. This aligns with the paper’s results but differs methodologically, avoiding the joint Koszul-complex framework. Minor gaps in the model’s write-up (e.g., explicit justification that every joint resonant state lies in a generalized X_{A0}-resonant space, and the general half-space bound) are standard and addressable using known one-parameter microlocal theory; the paper covers these points via its intrinsic/cohomological construction (e.g., Proposition 4.9 and Corollary 4.16) .