SLICES OF MULTIFRACTAL MEASURES AND APPLICATIONS TO RAINFALL DISTRIBUTIONS

We discuss the relationship between the multifractal functions of a plane measure and those of slices or sections of the measure with a line. Motivated by recent mathematical ideas about the relationship between measures and their slices, we formulate the "slice hypothesis," and consider the theoretical limitations of this hypothesis. We compute the multifractal functions of several standard self-similar and self-affine measures and their slices to examine the validity of the slice hypothesis. We are particularly interested in using the slice hypothesis to estimate multifractal properties of spatial rainfall fields by analyzing rainfall data representing slices of rainfall fields. We consider how rainfall time series at a fixed site and slices of composite radar images can be used for this purpose, testing this on field data from a radar composite in the USA and on appropriate time series.

Entire functions with Julia sets of zero measure

We extend results of McMullen about the dynamics of entire functions for which the orbits of the critical values stay away from the Julia set. In particular we show that such functions are expanding on their Julia sets which have self-similarity properties. Under suitable further conditions the Julia sets have plane measure zero.

A Set of Plane Measure Zero Containing all Finite Polygonal Arcs

We say a (plane) set A contains all sets of some type if, for each B of type , there is a subset of A that is congruent to B. Recently, Besicovitch and Rado [3] and independently, Kinney [5] have constructed sets of plane measure zero containing all circles. In these papers it is pointed out that the set of all similar rectangles, some sets of confocal conies and other such classes of sets can be contained in sets of plane measure zero, but all these generalizations rely in some way on the symmetry, or similarity of the sets within the given type.In this paper we construct a set of plane measure zero containing all finite polygonal arcs (i.e., the one-dimensional boundaries of all polygons with a finite number of sides) with slightly stronger results if we restrict our attention to k-gons for some fixed k.