Long-time behavior of the semiclassical baker’s map

We study the long-time behavior of the quantized baker’s map in the semi-classical approximation. Our main object of investigation is the trace of the (n-step) time-evolution operator. We express this trace as a phase-space integral which equals the semiclassical expression in terms of periodic orbits. This enables us to follow the evolution explicitly up to the time at which the semiclassical traces start to diverge exponentially from the quantum ones. Our data indicate that this breakdown time scales with h in a way close to ${\mathit{h}}^{\mathrm{-}1/2}$.