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Under certain hypothesis on the underlying classical Hamiltonian flow, weproduce local scaling asymptotics in the semiclassical regime for aBerezin-T\"oplitz version of the Gutzwiller trace formula on a quantizablecompact K\"ahler manifold, in the spirit of the near-diagonal scalingasymptotics of Szeg\"o and T\"oplitz kernels. More precisely, we consider ananalogue of the \lq Gutzwiller-T\"oplitz kernel\rq\, previously introduced inthis setting by Borthwick, Paul and Uribe, and study how it asymptoticallyconcentrates along the appropriate classical loci defined by the dynamics, withan explicit description of the exponential decay along normal directions. Theselocal scaling asymptotics probe into the concentration behavior of theeigenfunctions of the quantized Hamiltonian flow. When globally integrated,they yield the analogue of the Gutzwiller trace formula.

Local scaling asymptotics for the Gutzwiller trace formula in Berezin-T\"oplitz quantization