Dynamics of injective quasi-contractions

Abstract

We study injective locally contracting maps of the interval. After giving an upper-bound on the number of ergodic components, we show that generically finitely many periodic orbits attract the whole dynamics and that this picture is stable under perturbation. In relation to the problem of maximizing measures for regular maps, we then consider a class of probability measures on the circle invariant by × p generalizing the family of Sturm measures and show its generic periodic character. We detail the structure of order-preserving locally contracting maps on the circle. The rotation number is shown to be generically rational and the transformations having a given rational rotation number are shown explicitly. We also count the periodic attractors. We then deduce the existence of measurable conjugacies with three-interval exchange transformations in non-periodic cases for a model with three pieces on the interval.