Israel Journal of Mathematics

, 106:313 | Cite as

Linear expansions, strictly ergodic homogeneous cocycles and fractals

  • Teturo Kamae


We consider a compact space Θ on whichR acts additively andR + acts multiplicatively satisfying the distributive law. Moreover,R-action is strictly ergodic. Such Θ is constructed as a space of colored tilings corresponding to a weighted substitution, which is a kind of natural extension of thef-expansion for a piecewise linearf. We define a homogeneous cocycleF on Θ, which was called a cocycle with the scaling property in [2]. This is a realization of fractal functions which admit the continuous scalings. This also defines a self-similar process with strictly ergodic, stationary increments which has 0 entropy.


Metrizable Space Stationary Increment Compact Metrizable Space Multiplicative Subgroup Colored Tiling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [1]
    T. Bedford and T. Kamae,Stieltjes integration and stochastic calculus with respect to self-affine functions, Japan Journal of Industrial and Applied Mathematics8 (1991), 445–459.MATHMathSciNetGoogle Scholar
  2. [2]
    J-M. Dumont, T. Kamae and S. Takahashi,Minimal cocycles with the scaling property and substitutions, Israel Journal of Mathematics95 (1996), 393–410.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    W. Feller,An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York, 1966.MATHGoogle Scholar
  4. [4]
    T. Kamae and M. Keane,A class of deterministic self-affine processes, Japan Journal of Applied Mathematics7 (1990), 185–195.MathSciNetCrossRefGoogle Scholar
  5. [5]
    T. Kamae and S. Takahashi,Ergodic Theory and Fractals, Springer-Verlag, Tokyo, 1993 (in Japanese).Google Scholar

Copyright information

© Hebrew University 1998

Authors and Affiliations

  1. 1.Department of MathematicsOsaka City UniversityJapan

Personalised recommendations