Kenneth Falconer

The Hausdorff dimension of self-affine fractals

If T is a linear transformation on ℝ n with singular values α 1 ≥ α 2 ≥ … ≥ α n , the singular value function o s is defined by where m is the smallest integer greater than or equal to s . Let T 1 , …, T k be contractive linear transformations on ℝ n . Let where the sum is over all finite sequences (i 1 , …, i r ) with 1 ≤ i j ≤ k. Then for almost all (a 1 , …, a k ) ∈ ℝ nk , the unique non-empty compact set F satisfying has Hausdorff dimension min{ d, n }. Moreover the ‘box counting’ dimension of F is almost surely equal to this number.

The multifractal spectrum of statistically self-similar measures

We calculate the multifractal spectrum of a random measure constructed using a statistically self-similar process. We show that with probability one there is a multifractal decomposition analogous to that in the deterministic self-similar case, with the exponents given by the solution of an expectation equation.

Dimensions and measures of quasi self-similar sets

We show that sets with certain quasi self-similar properties have equal Hausdorff and box-packing dimensions and also have positive and finite Hausdorff measure at the dimensional value. A number of applications of these results to particular examples are given.

The dimension of self-affine fractals II

A family { S 1 , , S k } of contracting affine transformations on R n defines a unique non-empty compact set F satisfying . We obtain estimates for the Hausdorff and box-counting dimensions of such sets, and in particular derive an exact expression for the box-counting dimension in certain cases. These estimates are given in terms of the singular value functions of affine transformations associated with the S i . This paper is a sequel to 4, which presented a formula for the dimensions that was valid in almost all cases.

A subadditive thermodynamic formalism for mixing repellers

The thermodynamical description of fractals that has recently attracted much interest both experimentally and theoretically in the study of dynamical systems is, in some ways, limited, being essentially an additive theory. The author presents a subadditive thermodynamic formalism for which he derives a variational principle and shows how it may be used to study the dynamics of non-conformal transformations. In particular the author discusses an analogue of Bowens formula for the dimension of a mixing repeller.

Fractal Geometry: Mathematical Foundations and Applications.

Part I Foundations: mathematical background Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractals products of fractals intersections of fractals. Part II Applications and examples: fractals defined by transformations examples from number theory graphs of functions examples from pure mathematics dynamical systems iteration of complex functions-Julia sets random fractals Brownian motion and Brownian surfaces multifractal measures physical applications.

On the geometry of random cantor sets and fractal percolation

Random Cantor sets are constructions which generalize the classical Cantor set, “middle third deletion” being replaced by a random substitution in an arbitrary number of dimensions. Two results are presented here. (a) We establish a necessary and sufficient condition for the projection of ad-dimensional random Cantor set in [0,1]d onto ane-dimensional coordinate subspace to contain ane-dimensional ball with positive probability. The same condition applies to the event that the projection is the entiree-dimensional unit cube [0,1]e. This answers a question of Dekking and Meester,(9) (b) The special case of “fractal percolation” arises when the substitution is as follows: The cube [0,1]d is divided intoMd subcubes of side-lengthM−, and each such cube is retained with probabilityp independently of all other subcubes. We show that the critical valuepc(M, d) ofp, marking the existence of crossings of [0,1]d contained in the limit set, satisfiespc(M, d)→pc(d) asM→∞, wherepc(d) is the critical probability of site percolation on a latticeLd obtained by adding certain edges to the hypercubic lattice ℤd. This result generalizes in an unexpected way a finding of Chayes and Chayes,(4) who studied the special case whend=2.

Tangent Fields and the Local Structure of Random Fields

A tangent field of a random field X on ℝN at a point z is defined to be the limit of a sequence of scaled enlargements of X about z. This paper develops general properties of tangent fields, emphasising their rich structure and strong invariance properties which place considerable constraints on their form. The theory is illustrated by a variety of examples, both of a smooth and fractal nature.

Packing dimensions of projections and dimension profiles

For E a subset of ℝ n and 0 [les ] m [les ] n we define a ‘family of dimensions’ Dim m E , closely related to the packing dimension of E , with the property that the orthogonal projection of E onto almost all m -dimensional subspaces has packing dimension Dim m E . In particular the packing dimension of almost all such projections must be equal. We obtain similar results for the packing dimension of the projections of measures. We are led to think of Dim m E for m ∈ [0, n ] as a ‘dimension profile’ that reflects a variety of geometrical properties of E , and we characterize the dimension profiles that are obtainable in this way.

Multifractional, multistable, and other processes with prescribed local form

We present a general method for constructing stochastic processes with prescribed local form, encompassing examples such as variable amplitude multifractional Brownian and multifractional α-stable processes. We apply the method to Poisson sums to construct 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.