The limit set for piecewise linear contractions

Inspired by a voting model by Edward Phragmén, we prove some basic results for a dynamical system given by a piecewise linear and contractive map f(x)= the fractional part of (ax+b), where a,b<1, on the unit interval that takes two possible values at the point of discontinuity. We prove that there exists a universal limit cycle in the non-exceptional cases (in the exceptional case, the attractor is a Cantor set), and that the exceptional parameter set is very tiny in terms of gauge functions. The exceptional two-dimensional parameter (a,b) is shown to have Hausdorff-dimension one. This is joint work with Svante Janson