# Semi-classical quantum maps of semi-hyperbolic type

## TL;DR

This paper investigates semi-classical quantum maps associated with semi-hyperbolic periodic orbits, constructing operators that account for nearby orbits and deriving quantization rules in this complex dynamical setting.

## Contribution

It introduces a construction of monodromy and Grushin operators for semi-hyperbolic orbits, extending previous methods to include nearby orbit families and refine quantization rules.

## Key findings

- Constructed monodromy and Grushin operators for semi-hyperbolic orbits.
- Extended previous quantization methods to account for orbit families near the periodic orbit.
- Compared new constructions with existing approaches, highlighting differences in orbit inclusion.

## Abstract

Let M = R n or possibly a Riemannian, non compact manifold. We consider semi-excited resonances for a h-differential operator H(x, hD x ; h) on L 2 (M) induced by a non-degenerate periodic orbit $\gamma$ 0 of semi-hyperbolic type, which is contained in the non critical energy surface {H 0 = 0}. By semi-hyperbolic, we mean that the linearized Poincar{\'e} map dP 0 associated with $\gamma$ 0 has at least one eigenvalue of modulus greater (or less) than 1, and one eigenvalue of modulus equal to 1, and by non-degenerate that 1 is not an eigenvalue, which implies a family $\gamma$(E) with the same properties. It is known that an infinite number of periodic orbits generally cluster near $\gamma$ 0 , with periods approximately multiples of its primitive period. We construct the monodromy and Grushin operator, adapting some arguments by [NoSjZw], [SjZw], and compare with those obtained in [LouRo], which ignore the additional orbits near $\gamma$ 0 , but still give the right quantization rule for the family $\gamma$(E).