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We study the behavior of an open quantum system, with an $N$--dimensionalspace of states, whose density matrix evolves according to a non--unitary mapdefined in two steps: A unitary step, where the system evolves with anevolution operator obtained by quantizing a classically chaotic map (baker'sand Harper's map are the two examples we consider). A non--unitary step wherethe evolution operator for the density matrix mimics the effect of diffusion inthe semiclassical (large $N$) limit. The process of decoherence and thetransition from quantum to classical behavior are analyzed in detail by meansof numerical and analitic tools. The existence of a regime where the entropygrows with a rate which is independent of the strength of the diffusioncoefficient is demonstrated. The nature of the processes that determine theproduction of entropy is analyzed.Comment: 16 pages, 11 figure

Decoherence for classically chaotic quantum maps