  It is well-known that the action of a hyperbolic element (``cat map'') of the
modular group on the 2-torus has strong chaotic dynamical properties such as
mixing and exponential decay of correlations. In this note we study stability
of this behaviour with respect to kicks. Our approach is based on geometric
group theory, and in particular on a new result on quasimorphisms of the
modular group.
