  We study dynamical properties of generalized Bowen-Series boundary maps
associated to cocompact torsion-free Fuchsian groups. These maps are defined on
the unit circle (the boundary of the Poincar\'e disk) by the generators of the
group and have a finite set of discontinuities. We study the two forward orbits
of each discontinuity point and show that for a family of such maps the cycle
property holds: the orbits coincide after finitely many steps. We also show
that for an open set of discontinuity points the associated two-dimensional
natural extension maps possess global attractors with finite rectangular
structure. These two properties belong to the list of "good" reduction
algorithms, equivalence or implications between which were suggested by Don
Zagier.
