Methods from Multiscale Theory and Wavelets Applied to Nonlinear Dynamics

We show how fundamental ideas from signal processing, multiscale theory and wavelets may be applied to nonlinear dynamics. The problems from dynamics include itereated function systems (IFS), dynamical systems based on substitution such as the discrete systems built on rational functions of one complex variable and the corresponding Julia sets, and state spaces of subshifts in symbolic dynamics. Our paper serves to motivate and survey our recent results in this general area. Hence we leave out some proofs, but instead add a number of intuitive ideas which we hope will make the subject more accessible to researchers in operator theory and systems theory. Mathematics Subject Classification (2000). 42C40, 42A16, 43A65, 42A65.