Morse Filtrations and Heteroclinic Fusion of Pollicott–Ruelle Resonances

We consider Pollicott–Ruelle resonances for open hyperbolic Axiom A flows with finitely many recurrent basic pieces and no heteroclinic cycles. Near an arbitrary resonance, a finite recurrent trace realization gives the local Keldysh root module of the Dyatlov–Guillarmou Fredholm family a filtration by the lower ideals of the Morse graph. The associated graded module is the direct sum of the component root modules, while the off-diagonal heteroclinic blocks define successive, flag-relative Yoneda classes. Consequently the Fredholm divisor is additive even when the root module is nonsplit. We identify the extension class by a matrix quotient over the local ring of holomorphic germs and give a left–right null-vector criterion for the fusion of two simple component resonances into a length-two chain. The recurrent realization is obtained from a finite trace network with smooth virtual cuts, anisotropic gluing, and holomorphic Schur elimination. It is stably equivalent both to the intrinsic open-flow problem and to positive terminal absorption. The unfiltered root module and its Smith elementary divisors are intrinsic; for a fixed Morse lower-ideal flag, the extension classes are preserved by filtration-preserving changes of realization. We construct a smooth non-arithmetic graph-directed Axiom A suspension with two mixing basic pieces and a positive one-way heteroclinic channel. A nonnegative return potential normalizes the stable-Jacobian ground multiplier so that the bridge pairing is nonzero and equals the off-diagonal Grushin coefficient. The resulting global root module is Oz0/((z - z0)^2).