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2018 | OriginalPaper | Chapter

1. Introduction

Author : Viviane Baladi

Published in: Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

Publisher: Springer International Publishing

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Abstract

In this chapter, we first define the transfer operator, dynamical determinants, and the spectral and determinantal resonances of a weighted differentiable dynamical system. We then state the main results linking these resonances. We also discuss anisotropic spaces and the techniques used to prove these results, illustrating them by simple examples.

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next chapter Smooth expanding maps: The spectrum of the transfer operator
Footnotes
1
We exclude the easier case where \(T\) is analytic.
 
2
The infimum over those positive real numbers \(\rho\) such that the spectrum of the operator outside of the disc of radius \(\rho\) consists of isolated eigenvalues of finite multiplicity. Appendix A.1.
 
3
All inclusions are continuous.
 
4
For nonuniformly hyperbolic dynamics, for example Collet–Eckmann logistic maps, it would be interesting to make Definition 1.1 compatible with the results of Keller–Nowicki [110]. See Problem 2.​43.
 
5
In the nuclear case, this last claim holds if \(\mathcal{Q}\) is \(2/3\)-nuclear.
 
6
See also Remark 3.​1 there.
 
7
In the analytic setting, it turns out that the flat traces and thus the dynamical determinant coincide with the traces and determinant of the nuclear operators involved.
 
8
For expanding maps \(T\), we have \(\lim_{n \to\infty} \frac{1}{n}\sum_{k=0}^{n-1} \varphi\circ T^{k}(x) =\int\varphi\, \mathrm{d}\mu \) for every continuous function \(\varphi\) and almost all \(x\in M\), where \(\mu\) is an ergodic \(T\)-invariant probability measure. In addition, \(\mu\) is absolutely continuous with respect to Lebesgue measure. [14]
 
9
The corresponding result for expanding maps is classical, see e.g. [14] and the references therein.
 
10
The residues of the poles depend on \(\varphi\) and \(\psi\). In particular, they can vanish for some (non-generic) \(\varphi\) and \(\psi\).
 
11
As this book was going to press, Jézéquel [100] introduced an “intermediate” finite-dimensional ancillary matrix description which has its advantages, for example the proof of Proposition 3.​15 becomes simpler.
 
12
The analytic setting will not be discussed in detail in this book.
 
13
As explained in the preface, Liverani and Tsujii [119, 120] had previously obtained suboptimal results on \(d_{T, g}(z)\).
 
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Metadata
Title
Introduction
Author
Viviane Baladi
Copyright Year
2018
Book
Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

Print ISBN: 978-3-319-77660-6

Electronic ISBN: 978-3-319-77661-3

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-77661-3_1

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