Coincidence and disparity of fractal dimensions

Abstract

We investigate the dimension and structure of four fractal families: inhomogeneous attractors, fractal projections, fractional Brownian images, and elliptical polynomial spirals. For each family, particular attention is given to the relationships between different notions of dimension. This may take the form of determining conditions for them to coincide, or, in the case they differ, calculating the spectrum of dimensions interpolating between them. Material for this thesis is drawn from the papers [6,7,8,9,10].

First, we develop the dimension theory of inhomogeneous attractors for non-linear and affine iterated function systems. In both cases, we find natural quantities that bound the upper box-counting dimension from above and identify sufficient conditions for these bounds to be obtained. Our work improves and unifies previous theorems on inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer's seminal results on homogeneous self-affine sets.

Second, we prove that the intermediate dimensions of the orthogonal projection of a Borel set 𝐸 ⸦ ℝⁿ onto a linear subspace 𝑉 are almost surely independent of the choice of subspace. Similar methods identify the almost sure value of the dimension of Borel sets under index-α fractional Brownian motion. Various applications are given, including a surprising result that relates the box dimension of the Hölder images of a set to the Hausdorff dimension of the preimages.

Finally, we investigate fractal aspects of elliptical polynomial spirals; that is, planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis, computing explicitly their intermediate, box-counting and Assouad-type dimensions. Relying on this, we bound the Hölder regularity of maps that deform one spiral into another, generalising the `winding problem’ of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents.