Tokyo Journal of Mathematics

Invariant Measures for a Class of Rational Transformations and Ergodic Properties

Hiroshi ISHITANI and Kensuke ISHITANI

Full-text: Open access


This paper is concerned with giving explicitly the invariant density for a class of rational transformations from the real line $\mathbf{R}$ into itself. We proved that the invariant density can be written in terms of the fixed point $z_{0}$ in $\mathbf{C} \setminus \mathbf{R}$ or in terms of the periodic point $z_{0}$ in $\mathbf{C} \setminus \mathbf{R}$ with period 2. The explicit form of the density allows us to obtain the ergodic properties of the transformation $R$.

Article information

Tokyo J. Math., Volume 30, Number 2 (2007), 325-341.

First available in Project Euclid: 4 February 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A05: Measure-preserving transformations
Secondary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 60F05: Central limit and other weak theorems


ISHITANI, Hiroshi; ISHITANI, Kensuke. Invariant Measures for a Class of Rational Transformations and Ergodic Properties. Tokyo J. Math. 30 (2007), no. 2, 325--341. doi:10.3836/tjm/1202136679.

Export citation


  • R. L. Adler and L. Flatto, Geodesic flows, interval maps and symbolic dynamics, Bull. Amer. Math. Soc., 25 (1991), 229–334.
  • R. Bowen, Bernoulli maps of the interval, Israel J. Math., 28 (1977), 161–168.
  • A. Boyarsky and P. Góra, Laws of chaos$:$ invariant measures and dynamical systems in one dimension, Birkhäuser: Boston, 1997.
  • F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Zeitschrift., 180 (1982), 119–140.
  • H. Ishitani, A central limit theorem of mixed type for a class of 1-dimensional transformations, Hiroshima Math. J., 16 (1986), 161–188.
  • 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.
  • A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481–488.
  • A. Lasota and J. A. Yorke, Exact dynamical systems and the Frobenius-Perron operator, Trans. Amer. Math. Soc., 273 (1982), 375–384.
  • T. Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc., 235 (1978), 183–192.
  • A. Rényi, Representation for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477–493.
  • 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.