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K. J. Falconer

Publications and source records attributed to K. J. Falconer.

9 recordsLinked to original sources
arXiv2018

A capacity approach to box and packing dimensions of projections and other images

Dimension profiles were introduced by Falconer and Howroyd to provide formulae for the box-counting and packing dimensions of the orthogonal projections of a set E or a measure on Euclidean space onto almost all m-dimensional subspaces. The original definitions of dimension profiles are somewhat awkward and not easy to work with. Here we rework this theory with an alternative definition of dimension profiles in terms of capacities of E with respect to certain kernels, and this leads to the box-counting dimensions of projections and other images of sets relatively easily. We also discuss other uses of the profiles, such as the information they give on exceptional sets of projections and dimensions of images under certain stochastic processes. We end by relating this approach to packing dimension.

math.MG↗
arXiv2018

Self-stabilizing processes based on random signs

A self-stabilizing processes $\{Z(t), t\in [t_0,t_1)\}$ is a random process which when localized, that is scaled to a fine limit near a given $t\in [t_0,t_1)$, has the distribution of an $α(Z(t))$-stable process, where $α: \mathbb{R}\to (0,2)$ is a given continuous function. Thus the stability index near $t$ depends on the value of the process at $t$. In an earlier paper we constructed self-stabilizing processes using sums over plane Poisson point processes in the case of $α: \mathbb{R}\to (0,1)$ which depended on the almost sure absolute convergence of the sums. Here we construct pure jump self-stabilizing processes when $α$ may take values greater than 1 when convergence may no longer be absolute. We do this in two stages, firstly by setting up a process based on a fixed point set but taking random signs of the summands, and then randomizing the point set to get a process with the desired local properties.

math.PR↗
arXiv2018

Self-stabilizing processes

We construct `self-stabilizing' processes {Z(t), t $\in [t_0,t_1)$}. These are random processes which when `localized', that is scaled around t to a fine limit, have the distribution of an $α$(Z(t))-stable process, where $α$ is some given function on R. Thus the stability index at t depends on the value of the process at t. Here we address the case where $α$: R $\to$ (0,1). We first construct deterministic functions which satisfy a kind of autoregressive property involving sums over a plane point set $Π$. Taking $Π$ to be a Poisson point process then defines a random pure jump process, which we show has the desired localized distributions.

math.PR↗
arXiv2011

Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure

For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known.

math.MG↗
arXiv2011

Local dimensions of measures on self-affine sets

We show that, in a generic setting, self-affine and almost self-affine measures are exact dimensional, with local dimension equal almost everywhere to the information dimension and given by the zero of a superadditive pressure functional.

math.MG↗
arXiv2009

Generalised dimensions of measures on almost self-affine sets

We establish a generic formula for the generalised q-dimensions of measures supported by almost self-affine sets, for all q>1. These q-dimensions may exhibit phase transitions as q varies. We first consider general measures and then specialise to Bernoulli and Gibbs measures. Our method involves estimating expectations of moment expressions in terms of `multienergy' integrals which we then bound using induction on families of trees.

math.MG↗
arXiv2009

Dixmier traces and coarse multifractal analysis

We show how multifractal properties of a measure supported by a fractal F contained in [0,1] may be expressed in terms of complementary intervals of F and thus in terms of spectral triples and the Dixmier trace of certain operators. For self-similar measures this leads to a noncommutative integral over $F$ equivalent to integration with respect to an auxilary multifractal measure.

math.CA↗
arXiv2008

Multifractional, multistable, and other processes with prescribed local form

We present a general method for constructing stochastic processes with prescribed local form. Such processes include variable amplitude multifractional Brownian motion, multifractional $α$-stable processes, and multistable processes, that is processes that are locally $α(t)$-stable but where the stability index $α(t)$ varies with $t$. In particular we construct multifractional multistable processes where both the local self-similarity and stability indices vary.

math.PR↗