ABSTRACT
Dissipative and Hamiltonian chaotic dynamics trace “strange” shapes in their phase space. Such “strange” shapes are fractals. A fractal is a complex self-similar geometric object. Its sophisticated features are maintained ad infinitum under changes of the spatial scale. In some cases, the self-similarity is perfect, like in the Cantor set and the Koch curve. In some others, the self-similarity is not perfect, like in the Mandelbrot set. In nature, some shapes are fractal-like. They are not genuine fractal because their statistical self-similarity is maintained within a limited range of spatial scales and not ad infinitum. A parameter that characterizes a fractal is its dimension. There are several methods for determining the dimension of a fractal. One is the Box Counting method. Another one is a pointwise determination that is suitable to characterize multifractals. A multifractal is an interwoven set of fractals of different dimensions. Examples of multifractals are the dendrites that can be generated, for instance, by viscous fingering in a Hele-Shaw cell. A fractal environment affects the processes that occur in it at both the microscopic and macroscopic level. The evidence of this action is the widespread presence of power laws in many disciplines.
