Stability of piecewise rotations and affine maps

We study the dynamics of non-ergodic piecewise linear maps under perturbati ons. We show that for the class of maps that can be conjugated to piecewise rotations, sets following the same codings change Hausdorff continuously u nder perturbations of the map. It follows that the size f of the set of poi nts that iterate arbitrarily close to discontinuities changes semicontinuou sly under perturbations. This implies f changes continuously on a dense G(d elta) set and this supports a stronger numerical result by Ashwin, Chambers and Petkov, that the measure of such a set changes continuously. The main tools used in the paper include properties of convexity and symbolic dynami cs. The scope of our work includes maps that appear in digital filters. A d igital filter is a type of signal processing algorithm (software) used for filtering a digital signal.