Transformations Which Preserve Cauchy Distributions and Their Ergodic Properties

Abstract

This paper is concerned with invariant densities for transformations on $\mathbb{R}$ which are the boundary restrictions of inner functions of the upper half plane. G. Letac [9] proved that if the corresponding inner function has a fixed point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ or a periodic point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ with period 2, then a Cauchy distribution $(1/\pi)\mathrm{Im}\left(1/(x-z_{0}) \right)$ is an invariant probability density for the transformation. Using Cauchy's integral formula, we give an easier proof of Letac's result. An easy sufficient condition for such transformations to be isomorphic to piecewise expanding transformations on an finite interval is given by the explicit form of the density. Transformations of the forms $\alpha x + \beta - \sum ^{n }_{k=1}b_{k}/(x-a_{k})$, \:$\alpha x-\sum ^{\infty }_{k=1}\left\{ b_{k}/(x-a_{k})+b_{k}/(x+a_{k}) \right\}$ and $\alpha x +\beta\tan x$ are studied as examples.