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Linear expansions, strictly ergodic homogeneous cocycles and fractals. (English) Zbl 0914.28014

Consider a compact set carrying a strictly ergodic flow (action of \(\mathbb R\)), and in addition carrying a ‘scaling’ action of \({\mathbb R}_{>0}\) that distributes over the flow. Here, such actions are exhibited as spaces of coloured tilings associated to a weighted substitution. A homogeneous cocycle is associated to the action, and it is shown that this is a realization of ‘fractal functions’ with continuous scalings. This cocycle is used to define another self-similar process with strictly ergodic stationary increments and zero entropy.
Remark: the citation [2] in the abstract is to J.-H. Dumont, T. Kamae and S. Takahashi [Isr. J. Math. 95, 393-410 (1996; Zbl 0866.54033)].

MSC:

28D15 General groups of measure-preserving transformations
28A80 Fractals
54H20 Topological dynamics (MSC2010)

Citations:

Zbl 0866.54033
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References:

[1] Bedford, T.; Kamae, T., Stieltjes integration and stochastic calculus with respect to self-affine functions, Japan Journal of Industrial and Applied Mathematics, 8, 445-459 (1991) · Zbl 0769.60047
[2] Dumont, J-M.; Kamae, T.; Takahashi, S., Minimal cocycles with the scaling property and substitutions, Israel Journal of Mathematics, 95, 393-410 (1996) · Zbl 0866.54033 · doi:10.1007/BF02761048
[3] Feller, W., An Introduction to Probability Theory and Its Applications (1966), New York: Wiley, New York · Zbl 0138.10207
[4] Kamae, T.; Keane, M., A class of deterministic self-affine processes, Japan Journal of Applied Mathematics, 7, 185-195 (1990) · Zbl 0719.60041 · doi:10.1007/BF03167840
[5] Kamae, T.; Takahashi, S., Ergodic Theory and Fractals (1993), Tokyo: Springer-Verlag, Tokyo
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