Summary
We consider an iterated function system (IFS) of one-to-one contractive maps on a compact metric space. We define the top of an IFS; define an associated symbolic dynamical system; present and explain a fast algorithm for computing the top; describe an example in one dimension with a rich history going back to work of A. Rényi [Representations for Real Numbers and Their Ergodic Properties, Acta Math. Acad. Sci. Hung., 8 (1957), pp. 477–493]; and we show how tops may be used to help to model and render synthetic pictures in applications in computer graphics.
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References
M. F. Barnsley, Fractals Everywhere, Academic Press, New York, NY, 1988.
M. F. Barnsley and S. Demko, Iterated Function Systems and the Global Construction of Fractals, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 399 (1985), pp. 243–275.
M. F. Barnsley and L. F. Barnsley, Fractal Transformations, in ”The Colours of Infinity: The Beauty and Power of Fractals”, by Ian Stewart et al., Published by Clear Books, London, (2004), pp.66–81.
M. F. Barnsley, Ergodic Theory, Fractal Tops and Colour Stealing, Lecture Notes, Australian National University, 2004.
M. F. Barnsley, Theory and Application of Fractal Tops (Full Version), Preprint, Australian National University, 2005.
M.F. Barnsley, J.E. Hutchinson, O. Steflo A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy (2003) to appear in Fractals journal.
J. Elton, An Ergodic Theorem for Iterated Maps, Ergodic Theory Dynam. Systems, 7 (1987), pp. 481–488.
J. E. Hutchinson, Fractals and Self-Similarity, Indiana. Univ. Math. J., 30 (1981), pp. 713–749.
N. Lu, Fractal Imaging, Academic Press, (1997).
W. Parry, Symbolic Dynamics and Transformations of the Unit Interval, Trans. Amer. Math. Soc., 122 (1966), 368–378.
A. Rényi, Representations for Real Numbers and Their Ergodic Properties, Acta Math. Acad. Sci. Hung., 8 (1957), pp. 477–493.
I. Werner, Ergodic Theorem for Contractive Markov Systems, Nonlinearity 17 (2004) 2303–2313
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© 2005 Springer-Verlag London Limited
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Barnsley, M. (2005). Theory and Applications of Fractal Tops. In: Lévy-Véhel, J., Lutton, E. (eds) Fractals in Engineering. Springer, London. https://doi.org/10.1007/1-84628-048-6_1
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DOI: https://doi.org/10.1007/1-84628-048-6_1
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