Abstract
Recurrent iterated function systems generalize iterated function systems as introduced by Barnsley and Demko [BD] in that a Markov chain (typically with some zeros in the transition probability matrix) is used to drive a system of maps w j : K→ K, j = 1, 2,…, N, where K is a complete metric space. It is proved that under “average contractivity,” a convergence and ergodic theorem obtains, which extends the results of Barnsley and Elton [BE]. It is also proved that a Collage Theorem is true, which generalizes the main result of Barnsley et al. [BEHL] and which broadens the class of images which can be encoded using iterated map techniques. The theory of fractal interpolation functions [B] is extended, and the fractal dimensions for certain attractors is derived, extending the technique of Hardin and Massopust [HM]. Applications to Julia set theory and to the study of the boundary of IFS attractors are presented.
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Barnsley, M.F., Elton, J.H., Hardin, D.P. (1989). Recurrent Iterated Function Systems. In: DeVore, R.A., Saff, E.B. (eds) Constructive Approximation. Constructive Approximation. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-6886-9_1
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DOI: https://doi.org/10.1007/978-1-4899-6886-9_1
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