Continuous evolution of functions and measures toward fixed points of contraction mappings [chapter]

Jerry L. Bona, Edward R. Vrscay
Fractals in Engineering  
Let T be a contraction mapping on an appropriate Banach space B(X). Then the evolution equation yt = T y − y can be used to produce a continuous evolution y(x, t) from an arbitrary initial condition y0 ∈ B(X) to the fixed point 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
more » ... (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.
doi:10.1007/1-84628-048-6_15 fatcat:raz37chk5vdtbo3mhkjx5xmuje