Fractal Weyl law for open quantum chaotic maps
  We study the semiclassical quantization of Poincar\'e maps arising in scattering problems with fractal hyperbolic trapped sets. The main application is the proof of a fractal Weyl upper bound for the number of resonances/scattering poles in small domains near the real axis. This result encompasses the case of several convex (hard) obstacles satisfying a no-eclipse condition. 
St\'ephane Nonnenmacher, Johannes Sjoestrand, Maciej Zworski
math.AP math.DS nlin.CD