Jonathan M. Fraser

Assouad type dimensions and homogeneity of fractals

We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural ‘dimension pair’. In particular, we compute these dimensions for certain classes of self-affine sets and quasiself-similar sets and study their relationships with other notions of dimension, such as the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity.

Sixty Years of Fractal Projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For a long time the paper attracted little attention. But over the past 30 years, Marstrand’s projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.

The horizon problem for prevalent surfaces

We investigate the box dimensions of the horizon of a fractal surface defined by a function f ∈ C[0,1] 2 . In particular we show that a prevalent surface satisfies the ‘horizon property’, namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most α ,f orα ∈[ 2,3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.

Arithmetic patches, weak tangents, and dimension

The first named author is supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and the second named author is supported by a PhD scholarship provided bythe School of Mathematics in the University of St Andrews

The visible part of plane self-similar sets

Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1, then dimH VθF = 1 for almost all θ, where dimH denotes Hausdorff dimension. We confirm this when F is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected.

Micromeasure distributions and applications for conformally generated fractals

We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconers distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.

The Assouad dimension of randomly generated fractals

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the \emph{almost sure} or the \emph{Baire typical} Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

On the dimensions of a family of overlapping self-affine carpets

The work of J.M.F. was supported by the EPSRC grant EP/J013560/1 whilst at Warwick and an EPSRC doctoral training grant whilst at St Andrews.

The Assouad dimensions of projections of planar sets

The first named author is supported by a Leverhulme Trust Research Fellowship and the second named author is supported by the Academy of Finland through the grant Restricted families of projections and connections to Kakeya type problems, grant number 274512.

Dimension and measure for typical random fractals

We define a random iterated function system (RIFS) to be a finite set of (deterministic) iterated function systems (IFSs) acting on the same metric space. For a given RIFS, there exists a continuum of random attractors corresponding to each sequence of deterministic IFSs. Much work has been done on computing the ‘almost sure’ dimensions of these random attractors. Here we compute the typical dimensions (in the sense of Baire) and observe that our results are in stark contrast to those obtained using the probabilistic approach. Furthermore, we examine the typical Hausdorff and packing measures of the random attractors and give examples to illustrate some of the strange phenomena that can occur. The only restriction we impose on the maps is that they are bi-Lipschitz and we obtain our dimension results without assuming any separation conditions.