Kamae, Teturo Linear expansions, strictly ergodic homogeneous cocycles and fractals. (English) Zbl 0914.28014 Isr. J. Math. 106, 313-337 (1998). 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)]. Reviewer: Thomas Ward (Norwich) Cited in 1 ReviewCited in 7 Documents MSC: 28D15 General groups of measure-preserving transformations 28A80 Fractals 54H20 Topological dynamics (MSC2010) Keywords:fractal functions; continuous scalings; strictly ergodic cocycles Citations:Zbl 0866.54033 PDFBibTeX XMLCite \textit{T. Kamae}, Isr. J. Math. 106, 313--337 (1998; Zbl 0914.28014) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.