# Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets

@inproceedings{Mauldin2003GraphDM, title={Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets}, author={R. Daniel Mauldin and Mariusz Urbanski}, year={2003} }

Introduction 1. Symbolic dynamics 3. Holder 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.

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## 267 Citations

Scaling functions for graph directed Markov systems

- Mathematics, Computer Science
- 2019

The scaling function associated to a graph directed Markov system is introduced, and it is shown that it is a Holder continuous function of the dual symbolic Cantor set, and proved that the scaling function is a complete invariant of the conjugacy between limit sets.

Real analyticity of Hausdorff dimension for higher dimensional hyperbolic graph directed Markov systems

- Mathematics
- 2008

In this paper we prove that the Hausdorff dimension function of the limit sets of strongly regular, hyperbolic, conformal graph directed Markov systems living in higher dimensional Euclidean spaces…

REAL ANALYTICITY OF HAUSDORFF DIMENSION FOR HIGHER DIMENSIONAL GRAPH DIRECTED MARKOV SYSTEMS

- Mathematics
- 2006

In this paper we prove that the Hausdorff dimension function of the limit sets of strongly regular, countable, conformal graph directed Markov systems living in higher dimensional Euclidean spaces R,…

Recurrence rates for loosely Markov dynamical systems

- MathematicsJournal of the Australian Mathematical Society
- 2007

Abstract The concept of loosely Markov dynamical systems is introduced. We show that for these systems the recurrence rates and pointwise dimensions coincide. The systems generated by hyperbolic…

Title : Exponential convergence for Markov systems

- Mathematics
- 2015

Markov operators arising from graph directed constructions of iterated function systems are considered. Exponential convergence to an invariant measure is proved.

Exponential Convergence For Markov Systems

- Mathematics
- 2015

Abstract Markov operators arising from graph directed constructions of iterated function systems are considered. Exponential convergence to an invariant measure is proved.

Geometry of Markov systems and codimension one foliations

- Mathematics
- 2008

We show that the theory of graph directed systems can be used to study exceptional minimal sets of some foliated manifolds. A C1−smooth embedding of a contractible parabolically Markov system into a…

Non-autonomous conformal graph directed Markov systems

- Mathematics
- 2017

The theory of non-autonomous graph directed Markov systems is introduced which is a generalization of the theory of conformal graph directedMarkov systems of Mauldin and Urba\'nski and provides an application to non-Autonomous conformal dynamics by estimating the Hausdorff dimension of the Julia set of non -autonomous affine perturbations of an elliptic function from below.

HIGHER-DIMENSIONAL MULTIFRACTAL VALUE SETS FOR CONFORMAL INFINITE GRAPH DIRECTED MARKOV SYSTEMS

- Mathematics, Computer Science
- 2007

The level sets in the higher dimensional multifractal formalism for infinite conformal graph directed Markov systems are described and all values with non-empty level sets are identified and their Hausdorff dimension determined.