Interpolating between Hausdorff and box dimension

TL;DR

This paper introduces a family of intermediate fractal dimensions that interpolate between Hausdorff and box dimensions, generalizes their definition, and explores their properties in various fractal sets and transformations.

Contribution

It generalizes intermediate dimensions, provides conditions for their realization, and computes them for complex fractals like Bedford-McMullen carpets.

Findings

Intermediate dimensions can differ from traditional ones for certain sets.

A necessary and sufficient condition for a function to be an intermediate dimension is established.

Explicit formulas for intermediate dimensions of self-affine carpets are derived.

Abstract

Hausdorff and box dimension are two familiar notions of fractal dimension. Box dimension can be larger than Hausdorff dimension, because in the definition of box dimension, all sets in the cover have the same diameter, but for Hausdorff dimension there is no such restriction. This thesis focuses on a family of dimensions parameterised by $\theta \in (0,1)$, called the intermediate dimensions, which are defined by requiring that $\mbox{diam}(U) \leq (\mbox{diam}(V))^{\theta}$ for all sets $U,V$ in the cover. We begin by generalising the intermediate dimensions to allow for greater refinement in how the relative sizes of the covering sets are restricted. These new dimensions can recover the interpolation between Hausdorff and box dimension for compact sets whose intermediate dimensions do not tend to the Hausdorff dimension as $\theta \to 0$. We also use a Moran set construction to prove a necessary and sufficient condition, in terms of Dini derivatives, for a given function to be realised as the intermediate dimensions of a set. We proceed to prove that the intermediate dimensions of limit sets of infinite conformal iterated function systems are given by the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This applies to sets defined using continued fraction expansions, and has applications to dimensions of projections, fractional Brownian images, and general H\"older images. Finally, we determine a formula for the intermediate dimensions of all self-affine Bedford-McMullen carpets. The functions display features not witnessed in previous examples, such as having countably many phase transitions. We deduce that two carpets have equal intermediate dimensions if and only if the multifractal spectra of the corresponding uniform Bernoulli measures coincide.