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Erschienen in:
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2018 | OriginalPaper | Buchkapitel

4. Anisotropic Banach spaces defined via cones

verfasst von : Viviane Baladi

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

Verlag: Springer International Publishing

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Abstract

The main result of this chapter is a bound on the essential spectral radius of a weighted transfer operator associated with a differentiable diffeomorphism on a hyperbolic basic set and a differentiable weight function. The operator acts on two scales of anisotropic spaces of distributions on the manifold defined using cones (in the cotangent space) adapted to the diffeomorphism.

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Vorheriges Kapitel Smooth expanding maps: Dynamical determinants
Nächstes Kapitel A variational formula for the essential spectral radius
Fußnoten
1
If \(g\) is \(C^{\alpha}\) for some \(\alpha\in(r-1, r]\), this does not produce stronger results, in contrast to the expanding situation in Chapters 2 and 3.
 
2
We do not assume here that \(\Lambda\) is transitive.
 
3
In [21], the assumption that \(W^{u}\) is smooth could be lifted, by considering the supremum of the Triebel norm over charts which, instead of trivialising the foliation, places it in an unstable cone. However, a bunching condition was required, and the construction was rather heavy.
 
4
The contrived definition of \(\lambda_{x}(T^{m})\) is useful on the extended bundle \(E^{s}(x)\).
 
5
The choice of extensions is not essential.
 
6
Recall that a hyperbolic basic set is transitive by definition.
 
7
\(F\) will correspond to \(T^{m}\) in charts. In Part I, the map \(F\) was an inverse branch of \(T^{m}\) in charts. Cf (4.2) compared to (2.​5).
 
8
Here “spectral” refers to the spectral decomposition of the finite Markov transition matrix appearing when using symbolic dynamics, see [39, 14].
 
9
Recall that we assumed that \(M\) is connected.
 
10
Cone systems were called polarizations in [28, 31].
 
11
We denote the transposed matrix of \(A\) by \(A^{tr}\). We view \(\mathbf{C}_{\omega,\pm}\), \(\mathbf{C}'_{\omega,\pm}\) as locally constant cone fields in the cotangent bundle \(T^{*}\mathbb{R}^{d}\), so that \(F\) acts on these cones via the transpose of \(DF\).
 
12
Cone-hyperbolicity only depends on the data \(\mathbf{C}_{+}\) and \(\mathbf{C}'_{-}\).
 
13
Recall (4.6)–(4.7) and see e.g. [105, Cor 6.4.8]. Injectivity of the extension follows from the Hadamard–Lévy theorem.
 
14
We view \(\mathbf{C}_{\omega ,\pm}\) as locally constant cone fields in the cotangent bundle \(T^{*} \mathbb{R}^{d}\), so that the conditions are expressed with respect to normal subspaces.
 
15
In the original definition of [28, App. A], the multiplication by \((1-\psi_{0})\) had been inadvertently omitted.
 
16
By Lemma 4.21, we can equivalently take the completion of \(C^{r}(\overline{V})\). In addition, the same comment as in the footnote to Definition 2.​11 applies here.
 
17
The extendability condition (4.35) holds, for example, if \(U\) is a small ball and \(F\) is close enough to its derivative on \(U\), using the Hadamard–Lévy theorem to get injectivity.
 
18
In fact, the cone \(\tilde{\mathbf{C}}_{-}\) will not play any role below.
 
19
See Footnote 28 of Chapter 2 for a slightly different argument, following [31, App C].
 
20
Recall the definition of a sub-partition of unity from (2.​37).
 
21
The anisotropic Leibniz inequality Lemma D.10 involves a \(C^{u}\) norm for some \(u>0\), which can be arbitrarily large for a partition of unity.
 
22
This was not allowed for (2.​56), but it is licit here because the intersection multiplicity is bounded since \(T\) is now a diffeomorphism: There is no complexity at the end.
 
23
The piecewise hyperbolic situation in [21] is more delicate, and the corresponding arguments are different there, replacing (4.28) and (4.29) by complexity constants. See Problem 5.​31.
 
24
We may take \(\tilde{g}\) arbitrarily close to \(|g|\).
 
25
Instead of Theorem 2.​9, we used the more sophisticated [50, Thm 9], where \(b\) depends on \(x\) and not only on \(\xi\) in [15].
 
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Metadaten
Titel
Anisotropic Banach spaces defined via cones
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_4