Infinite ergodic measures for transformations on R with a discontinuity and horizontal asymptotes

Abstract

In the space of transformations, the study of the existence of infinite σ-finite measures is rather scarce, are limited to a couple of examples, one of which is the Boole transformation, which preserves the Lebesgue measure and is ergodic with respect to this measure. In this paper we consider this type of transformations with two horizontal asymptotes, showing for a large family of four parameters that, under certain conditions, exhibits an infinite ergodic measure and that under other conditions to the parameter space that family exhibits an ergodic probability measure; in both cases the measure is equivalent to the Lebesgue measure.