Technically, there is no connection between the fields of dynamical systems and fractal geometry. Dynamics is the study of objects in motion such as iterative processes; fractals are geometric objects that are static images. However, it has become apparent in recent years that most chaotic regions for dynamical systems are fractals. Hence, in order to understand chaotic behavior completely, one must pause to understand the geometric structure of fractals. There are then three major surprises that result from the random iterative process. The first surprise is that the fate of the orbit is a distinct geometric image, not a random mess as one might expect. A fractal is a subset of Rn which is self-similar and whose fractal dimension exceeds its topological dimension. One of the most important properties of a fractal is called self-similarity. There are many fractals that may be constructed via variations on the theme of infinite removals.