Minimal cocycles with the scaling property and substitutions |
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Authors: | Jean-Marie Dumont Teturo Kamae Satoshi Takahashi |
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Affiliation: | (1) Laboratoire de Mathématiques Discrètes, U.P.R. 9016, Case 930, 163 avenue de Luminy, 13288 Marseille Cedex 9, France;(2) Department of Mathematics, Osaka City University, Sugimoto 3-3-138, Sumiyoshi-ku, 558 Osaka, Japan;(3) Department of Mathematics, Osaka University, Machikaneyama-cho 1-1, Toyonaka, 560 Osaka, Japan |
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Abstract: | ‘Fractal’ functions are formulated as a minimal cocycle on a topological dynamics which admits nontrivial scaling transformations. In this paper, it is proved that if in addition it admits a continuous family of scaling transformations, then itscapacity is not ino(N 2). We define minimal cocycles with nontrivial scaling transformations coming from substitutions on a finite alphabet which are proved to have capacityO(N), so that they admit only a discrete family of scaling transformations. We also construct one which has capacityO(N 2) and admit a continuous family of scaling transformations. Supported by C.N.R.S.(U.R.A 225); Partially supported by the Japan Society for the Promotion of Science. |
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