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The Stokes geometry of the quantum Hénon map in the horseshoe regime. (English) Zbl 1342.37047

Summary: The Stokes geometry for the propagator of the quantum Hénon map is studied in the light of recent developments of the exact WKB analysis, especially in the case where the Hénon map satisfies the horseshoe condition. Our analysis reveals that the birth and death of the WKB solutions caused by the Stokes phenomenon do not occur in a local but entirely global manner, reflecting topological nature encoded in the Stokes geometry. We derive an explicit general formula to enumerate the number of WKB solutions in the asymptotic region and obtain its growth rate, which is shown to be less than the topological entropy of the corresponding classical dynamics.

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory

Biographic References:

Aoki, Takashi
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