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The saddle point method is known to be an efficient technique to approximately evaluate integrals. A difficulty in applying the saddle point method in general is the fact that not all of the saddles point solutions necessarily contribute. The Stokes geometry describes all geometrical information to judge which saddles should appear in its evaluations and which should be dropped, thereby it tells us how to construct global asymptotic solutions from the local pieces.
Here we present an idea for developing bifurcation theory of the Stokes geometry in quantum maps whose classical counterpart exhibit chaos. A concrete recipe to give the complete Stokes geometry is presented for the quantum Henon map, in which some new ingredients absent in conventional theory of asymptotics are taken into account. A key strategy is to first establish the Stokes geometry in the horseshoe limist, in which the underlying classical dynamics is described by the binary symbolic dynamics and the corresponding Stokes geometry is also analytically tractable, and then to trace it as a function of the system parameter by focusing on bifurcation of the Stokes geometry. The approach exactly follows the pruning front theory of classical symbolic dynamics. |
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