Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets
The book is devoted to geometric and dynamic theory of limit sets generated by iterations of contracting conformal maps. A far reaching generalization of this concept developed in the book is called a graph directed Markov system (GDMS), a notion based on a directed multigraph and an associated incidence matrix, determining which edges may follow a given edge. For each edge, we then have a 1-to-1 contraction between two compact metric spaces. The theory presented in the book also covers many settings that do not fit into the scheme of conformal iterated systems. The book contains chapters on symbolic dynamics (this chapter is self-contained and can be read independently of the rest of the book), Hölder families of functions and F-conformal measures (with a connection to the thermodynamic formalism and the Perron-Frobenius theorem), conformal graph directed Markov systems (the central chapter, dealing both with basic and more refined geometric properties of limit sets), conformal iterated function systems (a study of the Radon-Nikodym derivative of an invariant measure with respect to a conformal measure), parabolic iterated function systems and the Haussdorff and packing measures. The two short Appendices deal with ergodic theory and geometric measure theory. Many issues and current research topics will be interesting for a broad community of researchers in the theory of dynamical systems.