An invitation to dimension interpolation

Abstract

A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions of \emph{fractal dimension}. However, even for the simplest examples, the different definitions of dimension may completely disagree about the answer. In this expository article I will examine this phenomenon and use it to discuss and motivate \emph{dimension interpolation}. Dimension interpolation views these classical notions as boundary points of continuous families of dimensions, thus transforming isolated numerical answers into a coherent geometric picture.

Figures (1)

• Figure 1: Interpolation in action: plots of the intermediate dimensions (left), and the Assouad spectrum (right) as functions of $\theta \in (0,1)$ for the simple example $X =\{1/n\}_{n \in \mathbb{N}}$.