2013年6月
Transformations Which Preserve Cauchy Distributions and Their Ergodic Properties
TOKYO JOURNAL OF MATHEMATICS
- 巻
- 36
- 号
- 1
- 開始ページ
- 177
- 終了ページ
- 193
- 記述言語
- 英語
- 掲載種別
- 研究論文(学術雑誌)
- DOI
- 10.3836/tjm/1374497518
- 出版者・発行元
- PUBLICATION COMMITTEE, TOKYO JOURNAL MATHEMATICS
This paper is concerned with invariant densities for transformations on 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 C\R or a periodic point z(0) in C\R with period 2, then a Cauchy distribution (1/pi)Im (1/(x - z(0))) 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 - Sigma(n)(k=1)b(k)/(x - a(k)), alpha x - Sigma(infinity)(k=1){b(k)/(x - a(k)) + b(k)/(x + a(k))} and alpha x + beta tan x are studied as examples.
- リンク情報
- ID情報
-
- DOI : 10.3836/tjm/1374497518
- ISSN : 0387-3870
- Web of Science ID : WOS:000322425000009