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Erschienen in:
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2018 | OriginalPaper | Buchkapitel

5. A variational formula for the essential spectral radius

verfasst von : Viviane Baladi

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

Verlag: Springer International Publishing

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Abstract

This chapter describes a third scale of anisotropic Banach spaces of distributions, for which the best known bounds on the essential spectral radius of the transfer operator are known, improving those given in Chapter 4. The last section implements the Gouëzel-Keller-Liverani perturbation theory for this third type of Banach spaces.

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Vorheriges Kapitel Anisotropic Banach spaces defined via cones
Nächstes Kapitel Dynamical determinants for smooth hyperbolic dynamics
Fußnoten
1
Beware that the functions \(|\det DT|_{E^{u}(T)}|\) and \(|\det DT|_{E^{s}(T)}\) are only Hölder in general.
 
2
Kitaev used the notation \(\rho^{t,-s}(\mathcal{L})\) for our \(\rho^{t,s}(T,g)\).
 
3
The claim there is about “regular mixed transfer operators”, but we may apply [112, Lemma 2] to the regular mixed transfer operator induced by \(T\) and \(g\), using local charts and partitions of unity.
 
4
Regular cone-hyperbolicity only depends on the data \(\mathbf{C}_{+}\) and \(\mathbf{C}'_{-}\).
 
5
In the application we shall take \(s<0<t\).
 
6
Recall Remark 5.11.
 
7
Similarly, in [28] we use the Young inequality (2.​77) in the proof of Lemmas 4.​26 for \(p=\infty\).
 
8
In particular, there is no need to worry about intersection multiplicity.
 
9
If \(r\) is large enough and \(T_{\epsilon}\in C^{N}([-1,1],C^{r}(M,M))\) for \(N\ge3\), it is possible to write down explicit formulas for derivatives of higher order.
 
10
As mentioned above, this is a crucial starting point.
 
11
The proof of Lemma 5.17 shows that the boundedness claims hold for \(\mathcal{B}_{0}\) even if \(t-N<0\).
 
12
An analogous claim for spaces \(W^{t,s}_{p,*}\) is proved in [24, (2.6)].
 
13
Applications to Sinai billiards, in the spirit of [60, 19], would be especially interesting in higher dimensions, where only a conditional exponential mixing result is known [33, 34], assuming complexity control. (In dimension two, the “one-step expansion” argument in [60] allows us to show that hyperbolicity dominates complexity.)
 
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Metadaten
Titel
A variational formula for the essential spectral radius
verfasst von
Viviane Baladi
Copyright-Jahr
2018
Buch
Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

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

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

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-77661-3_5

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