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June 2013 Transformations Which Preserve Cauchy Distributions and Their Ergodic Properties
Hiroshi ISHITANI
Tokyo J. Math. 36(1): 177-193 (June 2013). DOI: 10.3836/tjm/1374497518

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.

Citation

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Hiroshi ISHITANI. "Transformations Which Preserve Cauchy Distributions and Their Ergodic Properties." Tokyo J. Math. 36 (1) 177 - 193, June 2013. https://doi.org/10.3836/tjm/1374497518

Information

Published: June 2013
First available in Project Euclid: 22 July 2013

zbMATH: 1351.37003
MathSciNet: MR3112382
Digital Object Identifier: 10.3836/tjm/1374497518

Subjects:
Primary: 37A05
Secondary: 37A50, 60F05

Rights: Copyright © 2013 Publication Committee for the Tokyo Journal of Mathematics

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Vol.36 • No. 1 • June 2013
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