Description

Monograph on Graph Directed Markov Systems with backgound and research level material. Graph Directed Markov Systems are a powerful tool with many application areas, including dynamical systems. This text, by acknowledged experts in the field, includes the necessary background material as well as an extensive list of references suggesting further reading. Graph Directed Markov Systems are a powerful tool with many application areas, including dynamical systems. This text, by acknowledged experts in the field, includes the necessary background material as well as an extensive list of references suggesting further reading. 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 - a Graph Directed Markov System. These systems are very robust in that they apply to many settings that do not fit into the scheme of conformal iterated systems. The basic theory is laid out here and the authors have touched on many natural questions arising in its context. However, they also emphasise the many issues and current research topics which can be found in original papers. For example the detailed analysis of the structure of harmonic measures of limit sets, the examination of the doubling property of conformal measures, the extensive study of generalized polynomial like mapping or multifractal analysis of geometrically finite Kleinian groups. This book leads readers onto frontier research in the field, making it ideal for both established researchers and graduate students. Introduction; 1. Symbolic dynamics; 3. Hölder families of functions; 4. Conformal graph directed Markov systems; 5. Examples of graph directed Markov systems; 6. Conformal iterated function systems; 7. Dynamical rigidity of conformal iterated function systems; 8. Parabolic iterated function systems; 9. Parabolic systems: Hausdorff and packing measures. "The authors provide in this monograph a state-of-the-art account on the geometry and dynamics of limit sets arising from graph directed Markov systems (GDMS)."
Marc Kessebohmer, Mathematical Reviews