Morse-Quiver Resonance Control for Open Hyperbolic Flows

We develop a local Morse–quiver description of controlled Pollicott–Ruelle resonances in open Axiom A flows. For each isolated finite spectral cluster, a filtration-preserving Grushin reduction has the form E_{-+}(s) = D(s) - Gamma(s), where the diagonal vertex families D are the component Keldysh data and the strictly triangular radical Gamma records resolved heteroclinic transport. Lower Morse ideals filter the global root module, whose associated graded is the direct sum of the component modules. Forgetting this filtration and applying local length to the filtered K0 class gives the resonance divisor. The two admissible control directions are asymmetric. Recurrent controls deform the component vertex families and move the attained component pressure edges; they may also change heteroclinic amplitudes. Pure wandering controls are supported in a no-return region disjoint from the fixed recurrent isolating neighborhoods. When they preserve the Morse filtration and Fredholm setup, every component family remains unchanged. The full resonance divisor is then fixed in the common Fredholm half-plane, although the framed presentation coordinates move inside an isodivisor fibre. For a unique controlled bridge with constant flight integral ell, the bridge coefficient satisfies gamma(b) = exp(-ell b) gamma(0). At a water-filling collision the nonsplit length-two root module persists for every finite b, whereas the second Laurent coefficient, the nilpotent Riesz part, and the polynomial transient prefactor decay like exp(-ell b). The local pseudospectral radius is comparable to epsilon + (epsilon |gamma(b)|)^(1/2). For an arbitrary finite active heteroclinic DAG with aligned simple edges, the triangular part remains in the same Morse incidence radical. A path–jet formula determines every Laurent coefficient. If the zero-jet sum over the longest active paths is nonzero, the maximal pole order equals one plus the longest directed path length. Either a unique longest path or a canonical positive normalization guarantees this nonvanishing. Edge controls attenuate a selected envelope of zero-jet path weights, which has a path-simplex dual. Thus pure wandering transport changes the framed extension data and the quantitative nonnormal response while preserving the divisor in the common Fredholm half-plane.