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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.

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
Full Text: DOI Euclid
[1] J. Aaronson, Ergodic theory for inner functions of the upper half plane, Ann. Inst. Henri Poincare 14 (1978), 233-253. · Zbl 0378.28009
[2] R. Bowen, Bernoulli maps of the interval, Israel J. Math. 28 (1977), 161-168. · Zbl 0377.28010
[3] A. Boyarsky and P. Góra, Laws of chaos: invariant measures and dynamical systems in one dimension , Birkhäuser: Boston, 1997. · Zbl 0893.28013
[4] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Zeitschrift. 180 (1982), 119-140. · Zbl 0485.28016
[5] H. Ishitani, A central limit theorem of mixed type for a class of 1-dimensional transformations, Hiroshima Math. J. 16 (1986), 161-188. · Zbl 0658.60042
[6] H. Ishitani, Central limit theorems for the random iterations of 1-dimensional transformations, Dynamics of complex systems, Kokyuroku, RIMS, Kyoto Univ. 1404 (2004), 21-31.
[7] H. Ishitani and K. Ishitani, Invariant measures for a class of rational transformations and ergodic properties, Tokyo J. Math. 30 (2007), 325-341. · Zbl 1145.37001
[8] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. · Zbl 0298.28015
[9] G. Letac, Which Functions Preserve Cauchy Laws? Proc. Amer. Math. Soc. 67 (1977), 277-286. · Zbl 0376.28019
[10] T. Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc. 235 (1978), 183-192. · Zbl 0371.28017
[11] N. F. G. Martin, On finite Blaschke products whose restrictions to the unit circle are exact endomorphisms, Bull. London Math. Soc. 15 (1983), 343-348. · Zbl 0487.28017
[12] J. Rousseau-Egele, Une théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab. 11 (1983), 772-788. · Zbl 0518.60033
[13] M. Sato, Theory of hyperfunctions I, J. Fac. Sci. Univ. Tokyo, Sec. I 8 (1959), 139-193. · Zbl 0087.31402
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