  We study a two-parameter family of one-dimensional maps and related
(a,b)-continued fractions suggested for consideration by Don Zagier. We prove
that the associated natural extension maps have attractors with finite
rectangular structure for the entire parameter set except for a Cantor-like set
of one-dimensional Lebesgue zero measure that we completely describe. We show
that the structure of these attractors can be "computed" from the data (a,b),
and that for a dense open set of parameters the Reduction theory conjecture
holds, i.e. every point is mapped to the attractor after finitely many
iterations. We also show how this theory can be applied to the study of
invariant measures and ergodic properties of the associated Gauss-like maps.
