arXiv subjects

Jonathan M. Fraser

Publications and source records attributed to Jonathan M. Fraser.

97 records · Page 6Linked to original sources

arXiv2013

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 quasi-self-similar sets and study their relationships with other notions of dimension, like 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.

math.MG↗

arXiv2013

Inhomogeneous self-similar sets and box dimensions

We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more difficult problem of computing the lower box dimension. We give some non-trivial bounds and provide examples to show that lower box dimension behaves much more strangely than the upper box dimension, Hausdorff dimension and packing dimension.

math.MG↗

arXiv2012

Dimension and measure for typical random fractals

We consider the dimension and measure of typical attractors of random iterated function systems (RIFSs). We define a RIFS to be a finite set of (deterministic) iterated function systems (IFSs) acting on the same metric space and, for a given RIFS, we define 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 a number of 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.

math.MG↗

arXiv2011

On the packing dimension of box-like self-affine sets in the plane

We consider a class of planar self-affine sets which we call "box-like". A box-like self-affine set is the attractor of an iterated function system (IFS) of affine maps where the image of the unit square, [0,1]^2, under arbitrary compositions of the maps is a rectangle with sides parallel to the axes. This class contains the Bedford-McMullen carpets and the generalisations thereof considered by Lalley-Gatzouras, Barański and Feng-Wang as well as many other sets. In particular, we allow the mappings in the IFS to have non-trivial rotational and reflectional components. Assuming a rectangular open set condition, we compute the packing and box-counting dimensions by means of a pressure type formula based on the singular values of the maps.

math.MG↗

arXiv2011

The Hausdorff dimension of graphs of prevalent continuous functions

We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on $[0,1]^d$ for $d \in \mathbb{N}$ and use this to obtain results on the `horizon problem' for fractal surfaces. We begin with a survey of previous results on the dimension of a generic continuous function.

math.MG↗

arXiv2011

The horizon problem for prevalent surfaces

We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in C[0,1]^2 $. In particular we show that a prevalent surface satisfies the `horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most α, for α$\in$ [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.

math.MG↗

arXiv2010

The visible part of plane self-similar sets

Given a compact subset $F$ of $\mathbb{R}^2$, the visible part $V_θF$ 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 $\dim_H F \geq 1$ then $\dim_H V_θF = 1$ for almost all $θ$, where $\dim_H$ 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.

math.MG↗