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Erschienen in:
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2018 | OriginalPaper | Buchkapitel

6. Dynamical determinants for smooth hyperbolic dynamics

verfasst von : Viviane Baladi

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

Verlag: Springer International Publishing

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Abstract

As in Chapters 4 and 5, we consider a diffeomorphism on a hyperbolic basic set and a differentiable weight. In this chapter, we study the associated weighted dynamical determinant, giving a lower bound on the disc in which this determinant is analytic and where its zeroes admit a spectral interpretation. We apply the results obtained on the weighted dynamical determinant to study the dynamical zeta function.

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Vorheriges Kapitel A variational formula for the essential spectral radius
Nächstes Kapitel Two applications of anisotropic spaces
Fußnoten
1
Or an \(C^{r}\) Anosov diffeomorphism on a connected manifold \(M\), not necessarily transitive.
 
2
This is the same definition as (3.​4) in the expanding case, except that \(|\det(\mathrm{Id}-DT^{-m}(x))|\) there is replaced by \(|\det(\mathrm{Id}-DT^{m}(x))|\) here. This is because the transfer operator \(\mathcal{L}_{g}\) is now defined by forward composition with \(T\).
 
3
By Lemma A.3 and the perturbation results from §5.​3 one can construct examples of \(C^{r}\) hyperbolic diffeomorphisms with nontrivial resonances for \(r<\infty\).
 
4
See e.g. [178] or [105] for a definition of expansive and specification. Beware that expansive is not the same as expanding.
 
5
See (6.40) below for a more precise estimate.
 
6
As usual, if \(\inf|g||_{\Lambda}=0\) we approach \(|g|\) by non-vanishing functions. See Appendix B
 
7
The decomposition is independent of \(t\) and \(s\).
 
8
This choice is not essential in the proof of Proposition 6.9, but it will be important to prove Proposition 6.11.
 
9
See also [31, App C] and Footnote 28 of Chapter 4.
 
10
Or see Footnote 19.
 
11
See also [31, App C] and Footnote 28 of Chapter 4.
 
12
To write a formal proof involving charts, we may integrate by parts as many times as we like with respect to \(x\) in the relevant kernels, as in the second step of the proof of Lemma 6.9.
 
13
See also [31, App C] and Footnote 28 of Chapter 4.
 
14
Note that \(\mathrm{tr}\, \,\mathbb{T}_{x}^{m}\ne0\) only if \(x\in V\). If \(T^{m}(x)=x\) this implies \(x\in\Lambda\).
 
15
For the sake of comparison with [88], note that their transfer operator is defined by composing with \(T^{-1}\) so \(E^{s}\) there replaces \(E^{u}\) here.
 
16
See [73, 137] or [80].
 
17
Kitaev worked in the slightly more general setting of Mixed Transfer Operators.
 
18
In this respect, [68, Remark 2 after Thm 5] should be taken with a grain of salt.
 
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Metadaten
Titel
Dynamical determinants for smooth hyperbolic dynamics
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_6