Quantum resonances and regularity islands in quantum maps

We study analytically as well as numerically the dynamics of a quantum map near a quantum resonance of an order q. The map is embedded into a continuo us unitary transformation generated by a time-independent quasi-Hamiltonian . Such a Hamiltonian generates at the very point of the resonance a local g auge transformation described by the unitary unimodular group SU(q). The re sonant energy growth is attributed to the zero Liouville eigenmodes of the generator in the adjoint representation of the group while the nonzero mode s yield saturating with time contribution. Tn a vicinity of a given resonan ce, the quasi-Hamiltonian is then found in the form of power expansion with respect to the detuning from the resonance. The problem is related in this way to the motion along a circle in a (q(2) - 1)-component inhomogeneous " magnetic" field of a quantum particle with q intrinsic degrees of freedom d escribed by the SU(q) group. This motion is in parallel with the classical phase oscillations near a nonlinear resonance. The most important role is p layed by the resonances with the orders much smaller than the typical local ization length q much less than l. Such resonances master for exponentially long though finite times the motion in some domains around them. Explicit analytical solution is possible for a few lowest and strongest resonances.