  In order to study the behaviour of discrete dynamical systems under adiabatic
cyclic variations of their parameters, we consider discrete versions of
adiabatically-rotated rotators. Paralleling the studies in continuous systems,
we generalize the concept of geometric phase to discrete dynamics and
investigate its presence in these rotators. For the rotated sine circle map, we
demonstrate an analytical relationship between the geometric phase and the
rotation number of the system. For the discrete version of the rotated rotator
considered by Berry, the rotated standard map, we further explore this
connection as well as the role of the geometric phase at the onset of chaos.
Further into the chaotic regime, we show that the geometric phase is also
related to the diffusive behaviour of the dynamical variables and the Lyapunov
exponent.
