Abstract Mauldin/Urbanski The investigators will continue ongoing research involving measures of various kinds-Hausdorff, packing, conformal, Gibbs state, etc.; associated functions-capacities, pressure and associated dimensions. These notions will be studied from two interlaced viewpoints. One is that of various dynamical systems and the other is that of geometric measure theory as applied to recursively generated objects. These approaches naturally meld with one another and lead to some interesting mixtures of ideas. During the last few years we have developed a fairly extensive theory of the iteration of infinitely many uniformly contracting conformal maps(hyperbolic systems) and have made some applications of it to some well known problems. We would like to extend this theory to cover a more structured iteration of maps-whose governed by directed graphs or substitutions and random iterations. We also wish to apply our theory to geometric object obtained as limit sets of Kleinian groups or more generally a system of maps which are not uniformly contracting(parabolic systems). We intend to develop the tools necessary to apply our theory to continue obtaining new results about the geometric structure of sets of continued fractions and relationships to classical arithmetic density results. We also plan to develop an appropriate multifractal formalism for our systems. We want to use these tools to describe some aspects of the geometry of the Julia sets of Collett-Eckmann rational functions. Finally, we want to obtain some approximation theorems which should be useful for obtaining analogues of known section and projection theorems concerning Hausdorff measures and dimension to packing measures and dimension and other mixed geometric measures. The specific problems of this proposal arise from many contexts and their solutions will involve a mixture of techniques from various areas: complex analysis, dynamics, functional analysis-theory of positive operators of the Reulle-Perron-Frobeni us type(transfer operators), measure theory and probability, statistical physics-equilibrium states, and thermodynamic formalism. This work arises from problems occurring in several diverse areas-statistical physics, thermodynamics, the theory of turbulence, geometry, algorithms or recipes for generating various geometric objects which are subject to random errors and finally, to dynamical systems where the long term behavior of the system is governed by some seemingly strange object. Our work focuses on some common features of all of these systems. First, we intend to show how, in a natural way, to produce measures associated with each system. For some systems, these measures give us a means of measuring the long term behavior of the system. For some systems, it gives us a means of quantifying the fine scale geometric structure of the object produced or of stating what the geometric structure will be like on average. To each of these measures is associated a dimension, a number which might not be an integer-a fractal dimension. These dimension numbers can be computed or estimated by some basic formulas which we have developed. This is useful for applications. This part can be carried out with the aid of computer studies and thus is of direct interest for pratical reasons. We intend to continue to develop applicable formulas for determining these dimensions and from that how to produce the associated measures which yield so much information about the system.
