Open Conformal Systems and Perturbations of Transfer Operators

by Mark Pollicott, Mariusz Urbanski

Overview

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, rational functions and meromorphic maps.

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.

Product Details

ISBN-13:	9783319721781
Publisher:	Springer International Publishing
Publication date:	03/10/2018
Series:	Lecture Notes in Mathematics , #2206
Edition description:	1st ed. 2017
Pages:	204
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

1. Introduction.- 2. Singular Perturbations of Classical Original Perron–Frobenius Operators on Countable Alphabet Symbol Spaces.- 3. Symbol Escape Rates and the Survivor Set K(U_n).- 4. Escape Rates for Conformal GDMSs and IFSs.- 5. Applications: Escape Rates for Multimodal Maps
and One-Dimensional Complex Dynamics.