Lecture Notes in Mathematics

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

Authors: Mayer, Volker, Skorulski, Bartlomiej, Urbanski, Mariusz

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About this book

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.

Table of contents (1 chapters)

  • Introduction

    Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski

    Pages 1-4

Buy this book

eBook $34.99
price for USA (gross)
  • ISBN 978-3-642-23650-1
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.95
price for USA
  • ISBN 978-3-642-23649-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2036
Copyright
2011
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-23650-1
DOI
10.1007/978-3-642-23650-1
Softcover ISBN
978-3-642-23649-5
Series ISSN
0075-8434
Edition Number
1
Number of Pages
X, 112
Number of Illustrations and Tables
3 illustrations in colour
Topics