  We analyze simple models of classical chaotic open systems and of their
quantizations (open quantum maps on the torus). Our models are similar to
models recently studied in atomic and mesoscopic physics. They provide a
numerical confirmation of the fractal Weyl law for the density of quantum
resonances of such systems. The exponent in that law is related to the
dimension of the classical repeller (or trapped set) of the system. In a
simplified model, a rigorous argument gives the full resonance spectrum, which
satisfies the fractal Weyl law. For this model, we can also compute a quantity
characterizing the fluctuations of conductance through the system, namely the
shot noise power: the value we obtain is close to the prediction of random
matrix theory.
