Quantum and classical echoes in scattering systems described by simple Smale horseshoes

We explore the quantum scattering of systems classically described by binar y and other low-order Smale horseshoes, in a stage of development where the stable island associated with the inner periodic orbit is large, but chaos around this island is well developed. For short incoming pulses we nd peri odic echoes modulating an exponential decay over many periods. The period i s directly related to the development stage of the horseshoe. We exemplify our studies with a one-dimensional system periodically kicked in time and w e mention possible experiments.