Introduction

Summary

The geometric and dynamic theory of the limit set generated by the iteration of finitely many similarity maps satisfying the open set condition has been well developed for some time now. Over the past several years, the authors have in turn developed a technically more complicated geometric and dynamic theory of the limit set generated by the iteration of infinitely many uniformly contracting conformal maps, a (hyperbolic) conformal iterated function system. This theory allows one to analyze many more limit sets, for example sets of continued fractions with restricted entries. We recall and extend this theory in the later chapters. The main focus of this book is the exploration of the geometric and dynamic properties of a far reaching generalization of a conformal iterated function system called a graph directed Markov system (GDMS). These systems are very robust in that they apply to many settings that do not fit into the scheme of conformal iterated systems. While the basic theory is laid out here and we touch on many natural questions arising in its context, we emphasize that there are many issues and current research topics which we do not cover: for examples, the detailed analysis of the structure of harmonic measures of limit sets provided in [UZd], the examination of the doubling property of conformal measures performed in [MU6], the extensive study of generalized polynomial like mappings (see [U7] and [SU]), the multifractal analysis of geometrically finite Kleinian groups (see [KS]), and the connection to quantization dimension from engineering (see [LM] and [GL]). There are many research problems in this active area that remain unsolved.