Nonlinear rotations on a lattice - Wan Rozali, Wan Nur Fairuz Alwani; Vivaldi, Franco (Taylor and Francis, 2018)
We consider a prototypical two-parameter family of invertible maps of Z2, representing rotations with decreasing rotation number. These maps describe the dynamics inside the island chains of a piecewise afﬁne discrete twist map of the torus, in the limit of ﬁne discretisation. We prove that there is a set of full density of points which,depending of the parameter values,are either periodic or escape to inﬁnity. The proof is based on the analysis of an interval-exchange map over the integers, with inﬁnitely many intervals.