Componentwise Fractal Resonance Counting and Wandering-Dimension Inflation for Open Hyperbolic Flows

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 C^{beta_alpha}, then N_alpha^cum(E, Gamma; a) <= C_{epsilon,Gamma,B} E^{1+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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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-07-22
DOI
https://doi.org/10.5281/zenodo.21482120
Primary Topic
Mathematical Dynamics and Fractals
Type
preprint
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preprint

Componentwise Fractal Resonance Counting and Wandering-Dimension Inflation for Open Hyperbolic Flows

CHAO MA
Zenodo (CERN European Organization for Nuclear Research)
Mathematical Dynamics and Fractals
preprint

Componentwise Fractal Resonance Counting and Wandering-Dimension Inflation for Open Hyperbolic Flows

CHAO MA
preprint en

Abstract

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 C^{beta_alpha}, then N_alpha^cum(E, Gamma; a) <= C_{epsilon,Gamma,B} E^{1+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.

Zenodo (CERN European Organization for Nuclear Research)
Sustainable cities and communities
Mathematical Dynamics and Fractals
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Componentwise Fractal Resonance Counting and Wandering-Dimension Inflation for Open Hyperbolic Flows — CHAO MA · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS