Simultaneous Multifractal Analysis of the Branching and Visibility Measure on a Galton-Watson Tree

Abstract

On the boundary of a Galton-Watson tree one can define the visibility measure by splitting mass equally between the children of each vertex, and the branching measure by splitting unit mass equally between all vertices in the nth generation and then letting n go to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of a joint multifractal spectrum, i.e., we ask for the Hausdorff dimension of the boundary points which have an unusual local dimension for both these measures simultaneously. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure. MSC classification: 60J80, 28A80.Keywords: multifractal spectrum, two-parameter spectrum, mixed spectrum, two-dimensional multifractal analysis, random tree, self-similar fractal, branching process, percolation. MotivationMultifractal analysis provides a way of encapsulating complex information about the fractal nature of an object in a single curve, the multifractal spectrum. In this paper we show how a multifractal analysis can also offer deep insight into the relationship of two fractal objects in the form of a two-parameter multifractal spectrum. In particular we shall see that when one of the analysed measures fails to obey the 'multifractal formalism' such an analysis can lead to the discovery of new phenomena which are deeply rooted in the geometry of these measures.Our test case is the boundary of a Galton-Watson tree with nonzero offspring at every vertex. This set is, on the one hand, a familiar and well-studied object in probability and, on the other hand, it represents the symbolic dynamics of a class of self-similar random fractals and as such it is representative of the behaviour of a wider range of fractal objects with statistical self-similarity. Two natural measures can be defined on the boundary of a Galton-Watson tree, the visibility measure and the branching measure. Both have been studied separately from a multifractal point of view.The visibility measure is easily defined, by starting at the root of the tree with a unit mass and, recursively, at each vertex splitting it equally among the children. If the offspring distribution of the Galton-Watson tree is nondegenerate and satisfies some mild moment conditions, the visibility measure is multifractal and the multifractal formalism, see e.g. Falconer [11], applies, see Kinnison [16] for details. The branching measure represents the uniform measure on the boundary. It can be defined by taking the uniform distribution on the vertices in the nth generation and taking a limit as n → ∞. This measure is not multifractal if the offspring variable has zero probability of taking the value one, and otherwise it is only multifractal in a weaker sense. Only unusually large values of the upper local dimension are possible, and can be represented in a spectrum of hyp...