Heuristic investigation of chaotic map** producing fractal objects

Abstract

In the literature most examples on fractals are related to images produced by certain iterative processes. Here we will instead discuss how similar results may appear by map** the unit circle using different, somewhat unusual functions. In principle this will be achieved by choosing a series of the form Σf(n) exp(inϕ)/n ^s wheref(n) is a function which may depend on, e.g., the structure of the numbern. In some casesf(n) is even obtained by a suitable random process. Further, the parameter s usually satisfies 0 <s ≤ 1 withs = 1 in most cases. However, we also investigate how the fractal structure disappears by taking s > 1. In our examples we accept singular results in isolated points. Finally we have tried to determine the dimensions of the resulting fractal objects.