Invariant Measures for a Class of Rational Transformations and Ergodic Properties

Abstract

This paper is concerned with giving explicitly the invariant density for a class of rational transformations from the real line $\mathbf{R}$ into itself. We proved that the invariant density can be written in terms of the fixed point $z_{0}$ in $\mathbf{C} \setminus \mathbf{R}$ or in terms of the periodic point $z_{0}$ in $\mathbf{C} \setminus \mathbf{R}$ with period 2. The explicit form of the density allows us to obtain the ergodic properties of the transformation $R$.