  An iterative map of the unit disc in the complex plane (Appendix) is used to
explore certain aspects of selfdual, four dimensional gauge fields
(quasi)periodic in the Euclidean time. These fields are characterized by two
topological numbers and contain standard instantons and monopoles as different
limits. The iterations do not correspond directly to a discretized time
evolution of the gauge fields. They are implemented in an indirect fashion.
First, (t,r,\theta,\phi) being the standard coordinates, the (r,t) half plane
is mapped on the unit disc in an appropriate way. This provides an (r,t)
parametrization (Sec.1) of Z_0, the starting point of the iterations and makes
the iterates increasingly complex functions of r and t. These are then
incorporated as building blocks in the generating function of the fields
(Sec.2). We explain (starting in Sec.1 and at different stages) in what sense
and to what extent some remarkable features of our map (indicated in the title)
are thus carried over into the continuous time development of the fields.
Special features for quasiperiodicity are studied (Sec.3). Spinor solutions
(Sec.4) and propagators (Sec.5) are discussed from the point of view of the
mapping. Several possible generalizations are indicated (Sec.6). Some broader
topics are discussd in conclusion (Sec.7).
