Weyl Laws for Open Quantum Maps

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.