Weyl Laws for Open Quantum Maps

Abstract: We find Weyl upper bounds for the quantum open baker's map in the semiclassical limit. For the number of eigenvalues in an annulus, we derive the asymptotic upper bound $\mathcal O(N^\delta)$ where $\delta$ is the dimension of the trapped set of the baker's map and $(2 \pi N)^{-1}$ is the semiclassical parameter, which improves upon the previous result of $\mathcal O(N^{\delta + \epsilon})$. Furthermore, we derive a Weyl upper bound with explicit dependence on the inner radius of the annulus for quantum open baker's maps with Gevrey cutoffs.