Assouad Dimension and the Open Set Condition

Abstract

The Assouad dimension is a measure of the complexity of a fractal set similar to the box counting dimension, but with an additional scaling requirement. In this thesis, we generalize Moran's open set condition and introduce a notion called grid like which allows us to compute upper bounds and exact values for the Assouad dimension of certain fractal sets that arise as the attractors of self-similar iterated function systems. Then for an arbitrary fractal set A, we explore the question of whether the Assouad dimension of the set of differences A - A obeys any bound related to the Assouad dimension of A. This question is of interest, as infinite dimensional dynamical systems with attractors possessing sets of differences of finite Assouad dimension allow embeddings into finite dimensional spaces without losing the original dynamics. We find that even in very simple, natural examples, such a bound does not generally hold. This result demonstrates how a natural phenomenon with a simple underlying structure has the potential to be difficult to measure.