Summary
Let T be a contraction mapping on an appropriate Banach space B(X). Then the evolution equation y t =T y − y can be used to produce a continuous evolution y(x, t) from an arbitrary initial condition y 0 ∊ B(X) to the fixed point \(\bar y\) ∊ B(X) of T. This simple observation is applied in the context of iterated function systems (IFS). In particular, we consider (1) the Markov operator M (on a space of probability measures) associated with an N-map IFS with probabilities (IFSP) and (2) the fractal transform T (on functions in L 1(X), for example) associated with an N-map IFS with greyscale maps (IFSM), which is generally used to perform fractal image coding. In all cases, the evolution equation takes the form of a nonlocal differential equation.
Such an evolution equation technique can also be applied to complex analytic mappings which are not strictly contractive but which possess invariant attractor sets. A few simple cases are discussed, including Newton's method in the complex plane.
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© 2005 Springer-Verlag London Limited
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Bona, J.L., Vrscay, E.R. (2005). Continuous evolution of functions and measures toward fixed points of contraction mappings. In: Lévy-Véhel, J., Lutton, E. (eds) Fractals in Engineering. Springer, London. https://doi.org/10.1007/1-84628-048-6_15
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DOI: https://doi.org/10.1007/1-84628-048-6_15
Publisher Name: Springer, London
Print ISBN: 978-1-84628-047-4
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