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Transformations which preserve Cauchy distributions and their ergodic properties. (English) Zbl 1351.37003
Summary: 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 [Proc. Am. Math. Soc. 67, 277–286 (1978; Zbl 0376.28019)] 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.

MSC:
 37A05 Dynamical aspects of measure-preserving transformations 37A50 Dynamical systems and their relations with probability theory and stochastic processes 60F05 Central limit and other weak theorems
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References:
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