Resonances in Hyperbolic Dynamics

The uploaded paper states and sketches the pressure-gap result: if the trapped set KE is a hyperbolic repeller with P(1/2)<0, then for any ε>0 and C>0 there are no resonances in R(E,Ch,(|P(1/2)|−ε)h) (Theorem 2) . Its proof outline uses a nonselfadjoint twist (complex deformation plus anisotropic weights), an Ehrenfest-time decomposition into words, a one-step hyperbolic dispersive gain ∼(Ju)−1/2, and thermodynamic formalism to control sums by e^{T(P(1/2)+ε)} . The model’s solution proves the same statement with a complex absorbing potential instead of complex deformation but follows the same core mechanism (weighted propagator, hyperbolic dispersion, Ehrenfest-time calculus, pressure control, Laplace transform for the resolvent). Differences are technical (CAP vs. complex deformation), not conceptual; the logical steps align with the paper’s sketch.