Absolutely Continuous Invariant Measures for a Class of Affine Interval Exchange Maps

Abstract

We consider a class $\mathcal {A}$ of affine interval exchange maps of the interval and we analyse several ergodic properties of the elements of this class, among them the existence of absolutely continuous invariant probability measures. The maps of this class are parametrised by two values a and b, where $a,b \in (0,1)$. There is a renormalization map T defined from $\mathcal {A}$ to itself producing an attractor given by the set $\mathcal {R}$ of pure rotations, i.e. the set of (a, b) such that $b = 1 - a$. The density of the absolutely continuous invariant probability and the rotation number of the elements of the class $\mathcal {A}$ are explicitly calculated. We also show how the continued fraction expansion of this rotation number can be obtained from the renormalization map.