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Open Conformal Systems and Perturbations of Transfer Operators

  • Mark Pollicott
  • Mariusz Urbański

Part of the Lecture Notes in Mathematics book series (LNM, volume 2206)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Mark Pollicott, Mariusz Urbański
    Pages 1-17
  3. Mark Pollicott, Mariusz Urbański
    Pages 53-85
  4. Mark Pollicott, Mariusz Urbański
    Pages 87-145
  5. Back Matter
    Pages 189-204

About this book

Introduction

The focus of this book is on open conformal dynamical systems corresponding to the escape of a point through an open Euclidean ball. The ultimate goal is to understand the asymptotic behavior of the escape rate as the radius of the ball tends to zero.  In the case of hyperbolic conformal systems this has been addressed by various authors. The conformal maps considered in this book are far more general, and the analysis correspondingly more involved.

The asymptotic existence of escape rates is proved and they are calculated in the context of (finite or infinite) countable alphabets, uniformly contracting conformal graph-directed Markov systems, and in particular, conformal countable alphabet iterated function systems. These results have direct applications to interval maps, meromorphic maps and rational functions.

Towards this goal the authors develop, on a purely symbolic level, a theory of singular perturbations of Perron--Frobenius (transfer) operators associated with countable alphabet subshifts of finite type and Hölder continuous summable potentials. This leads to a fairly full account of the structure of the corresponding open dynamical systems and their associated surviving sets.

Keywords

open dynamical systems conformal dynamical systems escape rates survivor sets thermodynamic formalism Perron-Frobenius (transfer) operator singular perturbations countable graph directed Markov systems and IFSs countable alphabet subshifts of finite type rational functions and interval maps

Authors and affiliations

  • Mark Pollicott
    • 1
  • Mariusz Urbański
    • 2
  1. 1.Department of MathematicsUniversity of WarwickCoventryUnited Kingdom
  2. 2.Department of MathematicsUniversity of North TexasDentonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-72179-8
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-72178-1
  • Online ISBN 978-3-319-72179-8
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site
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