Monotonicity properties of the family of trapezoidal maps

Trapezoidal maps which are everywhere expanding out of their plateau form a three parameter family T, up to affine changes of coordinates. We show that splitting T according to the various possible dynamical "behaviors" (we make this word precise in the process), yields a codimension one foliation. Some consequences of our result in terms of the monotonicity along simple one parameter families in T are then drawn. All together, aperiodic behavior is rare both from the topological and the measure theoretical point of view in T. © 1991 Springer-Verlag.