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Open Conformal Systems and Perturbations of Transfer Operators

  • Contains entirely original results which cannot be found elsewhere in the literature

  • Treats topics which are now the subject of rapidly developing extensive research

  • Serves both as a reference and as a source of inspiration for further original work


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


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.


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

  1. 1.Department of MathematicsUniversity of WarwickCoventryUnited Kingdom
  2. 2.Department of MathematicsUniversity of North TexasDentonUSA

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