Examines resonance counts in open hyperbolic flows, suggesting bounds for dynamics.
We prove componentwise upper bounds for Pollicott–Ruelle resonance counts in open hyperbolic flows. If a component trapped transversal has upper Minkowski dimension kappa_alpha and its invariant covector graph is Cbeta_alpha, then N_alpha^cum(E, Gamma; a) <= Cepsilon,Gamma,B E1+kappa_alpha/(1+beta_alpha)+epsilon uniformly on compact smooth control families. The proof combines an exact recurrent filtration of the intrinsic divisor with continuous Gaussian propagation, a finite-rank orthogonal Wilson absorber, and a signed-shell reduction on the matching trace space. We verify the criterion for smooth affine open suspensions with two recurrent horseshoes and a one-way wandering bridge. Reversing the bridge preserves the recurrent dynamics and full intrinsic divisor but changes the complete trapped-transversal dimension from 1 to 2 - 2delta. The componentwise packet exponent remains 3/2, whereas the unreduced whole-set exponent becomes 2 - delta.
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CHAO MA (2026) studied this question.
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