Linear expansions, strictly ergodic homogeneous cocycles and fractals
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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 . 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.
KeywordsMetrizable Space Stationary Increment Compact Metrizable Space Multiplicative Subgroup Colored Tiling
- T. Kamae and S. Takahashi,Ergodic Theory and Fractals, Springer-Verlag, Tokyo, 1993 (in Japanese).Google Scholar