Abstract
Using sofic systems, modifications of the self-similar sets of Hutchinson are defined as solutions of systems of fixed-point equations. Their Hausdorff dimension is determined.
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Bandt, C.: Self-similar sets 1. Markov shifts and mixed self-similar sets. Math. Nachr.142, 107–123 (1989).
Bandt, C.: Self-similar sets 4. Topology and measures. Proc. Conf. Topology and Measure V (Binz, GDR, 1987), pp. 8–16, Greifswald 1988.
Barnsley, M. F., Demko, S.: Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London A399, 243–275 (1985).
Barnsley, M. F., Elton, J. H.: A new class of Markov processes for image encoding. Adv. Appl. Prob.20, 14–32 (1988).
Bedford, T.: Dimension and dynamics for fractal recurrent sets. J. London Math. Soc. (2)33, 89–100 (1986).
Dekking, F. M.: Recurrent sets. Adv. in Math.44, 78–104 (1982).
Elton, J. H.: An ergodic theorem for iterated maps. Ergodic Theory Dynam. Sys.7, 481–488 (1987).
Falconer, K. J.: The Geometry of Fractal Sets. Cambridge: University Press. 1985.
Feiste, U.: A generalization of mixed invariant sets. Matematika35, 198–206 (1988).
Fischer, R.: Sofic systems and graphs. Mh. Math.80, 179–186 (1975).
Gilbert, W. J.: Complex bases and fractal similarity. Annales des Sciences Mathématiques du Québec11, 65–77 (1987).
Graf, S.: Statistically self-similar fractals. Probab. Theory Rel. Fields74, 357–392 (1987).
Hata, M.: On some properties of set-dynamical systems. Japan Acad.61, Ser. A, 99–102 (1985).
Hayashi, S.: Self-similar sets as Tarski's fixed points. Publ. Res. Inst. Math. Sci. Kyoto Univ.21, 1059–1066 (1985).
Hutchinson, J. E.: Fractals and self-similarity. Indiana Univ. Math. J.30, 713–747 (1981).
Kamae, T.: A characterization of self-affine functions. Japan J. Appl. Math.3, 271–280 (1986).
Mandelbrot, B. B.: The Fractal Geometry of Nature. San Francisco: Freeman. 1982.
Marion, J.: Mesures de Hausdorff et théorie de Perron-Frobenius des matrices non-negatives. Ann. Inst. Fourier Grenoble354, 99–125 (1985).
Mauldin, R. D., Williams, S. C.: Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc.304, 811–823 (1988).
Moran, P. A. P.: Additive funcions of intervals and Hausdorff measures. Proc. Cambridge Philos. Soc.42, 15–23 (1946).
Oppenheimer, P.: Real time design and animation of fractal plants and trees. Computer graphics20, 55–64 (1986).
Prusinkiewicz, P.: Graphical applications of L-systems. Proc. of Graphics Interface '86-Vision Interface '86, 247–253 (1986).
Schultz, M.: Hausdorff-Dimension von Cantormengen mit Anwendungen auf Attraktoren. Humboldt-Univ. Berlin: Dissertation. 1986.
Weiss, B.: Subshifts of finite type and sofic systems. Mh. Math.77, 462–474 (1973).
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Bandt, C. Self-similar sets 3. Constructions with sofic systems. Monatshefte für Mathematik 108, 89–102 (1989). https://doi.org/10.1007/BF01308664
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DOI: https://doi.org/10.1007/BF01308664