  We analyze the behaviour of two quantum dynamical entropies in connection
with the classical limit. Using strongly chaotic classical dynamical systems as
models (Arnold Cat Maps and Sawtooth Maps), we also propose a discretization
procedure that resembles quantization; even in this case, studies of quantum
dynamical entropy production are carried out and the connection with the
continuous limit is explored. In both case (quantization and discretization)
the entropy production converge to the Kolmogorov-Sinai invariant on
time-scales that are logarithmic in the quantization (discretization)
parameter.
