Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry

  • Volker Mayer
  • Mariusz Urbanski
  • Bartlomiej Skorulski

Part of the Lecture Notes in Mathematics book series (LNM, volume 2036)

Also part of the Ecole d'Eté Probabilit.Saint-Flour book sub series (volume 2036)

Table of contents

  1. Front Matter
    Pages i-x
  2. Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
    Pages 1-4
  3. Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
    Pages 5-15
  4. Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
    Pages 17-38
  5. Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
    Pages 39-45
  6. Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
    Pages 47-56
  7. Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
    Pages 57-68
  8. Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
    Pages 69-74
  9. Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
    Pages 75-91
  10. Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
    Pages 93-108
  11. Back Matter
    Pages 109-112

About this book

Introduction

The theory of random dynamical systems originated from stochastic
differential equations. It is intended to provide a framework and
techniques to describe and analyze the evolution of dynamical
systems when the input and output data are known only approximately, according to some probability distribution. The development of this field, in both the theory and applications, has gone in many directions. In this manuscript we introduce measurable expanding random dynamical systems, develop the thermodynamical formalism and establish, in particular, the exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen’s formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we arrive at a natural classification of the systems into two classes: quasi-deterministic systems, which share many
properties of deterministic ones; and essentially random systems, which are rather generic and never bi-Lipschitz equivalent to deterministic systems. We show that in the essentially random case the Hausdorff measure vanishes, which refutes a conjecture by Bogenschutz and Ochs. Lastly, we present applications of our results to various specific conformal random systems and positively answer a question posed by Bruck and Buger concerning the Hausdorff dimension of quadratic random Julia sets.

Keywords

37-XX Hausdorff dimension multifractal spectrum random dynamical systems thermodynamical formalism

Authors and affiliations

  • Volker Mayer
    • 1
  • Mariusz Urbanski
    • 2
  • Bartlomiej Skorulski
    • 3
  1. 1.Département de MathématiquesUniversité Lille 1Villeneuve d'AscqFrance
  2. 2.Department of MathematicsUniversity of North TexasDentonUSA
  3. 3.Departamento de MatematicasUniversidad Catolica del NorteAntofagastaChile

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-23650-1
  • Copyright Information Springer-Verlag Berlin Heidelberg 2011
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-23649-5
  • Online ISBN 978-3-642-23650-1
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book