Continuous evolution of functions and measures toward fixed points of contraction mappings

Abstract

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.