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Erschienen in:
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2018 | OriginalPaper | Buchkapitel

3. Smooth expanding maps: Dynamical determinants

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 variant of Ruelle’s theorem on the dynamical determinants of transfer operators associated with differentiable expanding dynamics and weights. The proof uses the Milnor-Thurston kneading operator approach. The contents of this chapter are a blueprint for the technically more involved situation of hyperbolic dynamics and the corresponding anisotropic Banach spaces in Part II.

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Vorheriges Kapitel Smooth expanding maps: The spectrum of the transfer operator
Nächstes Kapitel Anisotropic Banach spaces defined via cones
Fußnoten
1
A power of a formal power series is a formal power series, the exponential of a formal power series thus gives a formal power series, using the Taylor series at zero of the exponential function.
 
2
The case \(d=1\) was considered in the introduction.
 
3
In particular, the domain of analyticity obtained in Theorem 3.3 is not optimal for general nonlinear expanding dynamics.
 
4
If \(C>1\), different arguments will be used in Section 3.3 if \(\alpha>d+t\) and in Section 3.3.4 in the more difficult case of low differentiability \(\alpha\le d+t\).
 
5
If \(L>1\), different arguments will be used in Sections 3.3 and 3.3.4.
 
6
See the remark after Proposition 3.13.
 
7
A rate of convergence is given by \(\| \mathbf{I}_{\epsilon}(\varphi)-\varphi\|_{H^{t}_{p}(M)} \le C \epsilon^{t-t'} \|\varphi\|_{H^{t'}_{p}(M)} \) for all \(0\le t'< t\), see e.g. [25, Lemma 5.4].
 
8
The decomposition is independent of \(t\) and \(p\).
 
9
At the end of this section, we give an alternative proof, using regularised determinants, in which the kneading operator is explicited.
 
10
Problem 2.​46 would allow us to simplify the argument somewhat. Note also that a finite matrix of operators, indexed by \(\omega\), as in [100] can further streamline the proof without requiring a countable matrix as in [31]. These remarks also apply, for instance, to the proof of Proposition 3.18, and to hyperbolic settings.
 
11
As in the proof of Proposition 3.18, we can safely ignore the operators \(A_{t}\) there.
 
12
If \(\omega_{j}\ne \omega'(\overrightarrow{\omega}_{j})\), we may proceed as in the proof of Proposition 3.18, see (3.61).
 
13
As this book was going to press, M. Jézéquel [101] announced a series of new examples of non-polar singularities.
 
Literatur
5.
Artuso, R., Aurell, E., Cvitanović, P.: Recycling of strange sets. I. Cycle expansions. Nonlinearity 3, 325–359 (1990) MathSciNetCrossRef
6.
Artuso, R., Aurell, E., Cvitanović, P.: Recycling of strange sets. II. Applications. Nonlinearity 3, 361–386 (1990) MathSciNetCrossRef
7.
Atiyah, M.F., Bott, R.: Notes on the Lefschetz fixed point formula for elliptic complexes. Harvard University. Reprinted in Bott’s Collected Papers, Vol. 2 (1964)
8.
Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. I. Ann. of Math. 86, 374–407 (1967) MathSciNetCrossRef
10.
Baillif, M.: Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions. Duke Math. J. 124, 145–175 (2004) MathSciNetCrossRef
11.
Baillif, M., Baladi, V.: Kneading determinants and spectra of transfer operators in higher dimensions: the isotropic case. Ergodic Theory Dynam. Systems 25, 1437–1470 (2005) MathSciNetCrossRef
12.
Baladi, V.: Optimality of Ruelle’s bound for the domain of meromorphy of generalized zeta functions. Portugaliae Mathematica 49, 69–83 (1992) MathSciNetMATH
13.
20.
Baladi, V., Gouëzel, S.: Good Banach spaces for piecewise hyperbolic maps via interpolation. Annales de l’Institut Henri Poincaré/Analyse non linéaire 26, 1453–1481 (2009) MathSciNetCrossRef
21.
Baladi, V., Gouëzel, S.: Banach spaces for piecewise cone hyperbolic maps. J. Modern Dynam. 4, 91–135 (2010) MathSciNetMATH
22.
Baladi, V., Keller, G.: Zeta functions and transfer operators for piecewise monotone transformations. Comm. Math. Phys. 127, 459–477 (1990) MathSciNetCrossRef
23.
Baladi, V., Kitaev, A., Ruelle, D., Semmes, S.: Sharp determinants and kneading operators for holomorphic maps. Tr. Mat. Inst. Steklova 216, Din. Sist. i Smezhnye Vopr., 193–235 (1997); translation in Proc. Steklov Inst. Math. 216, 186–228 (1997)
25.
Baladi, V., Liverani, C.: Exponential decay of correlations for piecewise contact hyperbolic flows. Comm. Math. Phys. 314, 689–773 (2012) MathSciNetCrossRef
27.
30.
Baladi, V., Tsujii, M.: Dynamical determinants for hyperbolic diffeomorphisms via dyadic decomposition, unpublished manuscript (2005)
31.
Baladi, V., Tsujii, M.: Dynamical determinants and spectrum for hyperbolic diffeomorphisms. In: Burns, K., Dolgopyat, D., Pesin, Ya. (eds.) Probabilistic and Geometric Structures in Dynamics, pp. 29–68, Contemp. Math., 469, Amer. Math. Soc., Providence, RI (2008) CrossRef
42.
Butterley, O., Liverani, C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1, 301–322 (2007) MathSciNetCrossRef
43.
Butterley, O.: A Note on Operator Semigroups Associated to Chaotic Flows. Ergodic Theory Dynam. Systems 36, 1396–1408 (2016) (Corrigendum: 36, 1409–1410 (2016)) MathSciNetCrossRef
44.
Butterley, O., Eslami, P.: Exponential Mixing for Skew Products with Discontinuities. Trans. Amer. Math. Soc. 369, 783–803 (2017) MathSciNetCrossRef
46.
Buzzi, J, Keller, G.: Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps. Ergodic Theory Dynam. Systems 21, 689–716 (2001) MathSciNetMATH
53.
66.
Eslami, P.: Stretched-exponential mixing for \(C^{1+\alpha}\) skew products with discontinuities. Ergodic Theory Dynam. Systems 369, 783–803 (2017)
69.
Faure, F., Sjöstrand, J.: Upper bound on the density of Ruelle resonances for Anosov flows. Comm. Math. Phys. 308, 325–364 (2011) MathSciNetCrossRef
73.
74.
75.
Fried, D.: The flat-trace asymptotics of a uniform system of contractions. Ergodic Theory Dynam. Systems 15, 1061–1073 (1995) MathSciNetCrossRef
76.
Friedman, J.S.: The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations. Math. Z. 250, 939–965 (2005) MathSciNetCrossRef
78.
Gallavotti, G.: Funzioni zeta ed insiemi basilari. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 61, 309–317 (1977) MathSciNetMATH
81.
Gohberg, I., Goldberg, S., Krupnik, N.: Traces and Determinants of Linear Operators. Birkhäuser, Basel (2000) CrossRef
87.
Gouëzel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Ergodic Theory Dynam. Systems 26, 189–217 (2006) MathSciNetCrossRef
88.
Gouëzel, S., Liverani, C.: Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differential Geom., 79, 433–477 (2008) MathSciNetCrossRef
91.
Gundlach, V. M., Latushkin, Y.: A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces. Ergodic Theory Dynam. Systems 23, 175–191 (2003) MathSciNetCrossRef
98.
Hörmander, L.: The analysis of linear partial differential operators. III. Pseudo-differential operators. Grundlehren der Mathematischen Wissenschaften 274, Springer-Verlag, Berlin (Corrected reprint of the 1985 original, 1994) MATH
100.
Jézéquel, M.: Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response. arXiv:​1708.​01055
101.
Jézéquel, M.: Private communication (October 2017).
112.
Kitaev, A.Yu.: Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness. Nonlinearity 12, 141–179 (1999). Corrigendum: “Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness”. Nonlinearity 12, 1717–1719 (1999) MathSciNetCrossRef
119.
Liverani, C.: Fredholm determinants, Anosov maps and Ruelle resonances. Discrete Contin. Dyn. Syst. 13, 1203–1215 (2005) MathSciNetCrossRef
120.
122.
Mayer, D.: The Ruelle–Araki Transfer Operator in Classical Statistical Mechanics. Lecture Notes in Phys. 123, Springer-Verlag, Berlin-New York (1980) MATH
123.
Milnor J., Thurston W.: Iterated maps of the interval. In: Alexander J.C., ed., Dynamical Systems (Maryland 1986–1987), pp. 465–563, Lecture Notes in Math. 1342, Springer-Verlag, Berlin (1988)
125.
Nakano, Y., Tsujii, M., Wittsten, J.: The partial captivity condition for U(1) extensions of expanding maps on the circle. Nonlinearity 29, 1917–1925 (2016) MathSciNetCrossRef
126.
Naud, F.: Entropy and decay of correlations for real analytic semi-flows. Ann. Henri Poincaré. 10, 429–451 (2009) MathSciNetCrossRef
132.
Pietsch, A.: Eigenvalues and \(s\)-numbers. Cambridge Studies in Advanced Mathematics, 13. Cambridge University Press, Cambridge (1987) MATH
134.
Pollicott, M.: A complex Ruelle–Perron–Frobenius theorem and two counterexamples. Ergodic Theory Dynam. Systems 4, 135–146 (1984) MathSciNetCrossRef
137.
142.
Ruelle, D.: An extension of the theory of Fredholm determinants. Inst. Hautes Études Sci. Publ. Math. 72, 175–193 (1990) MathSciNetCrossRef
144.
Ruelle, D.: Dynamical zeta functions: where do they come from and what are they good for? in: Mathematical physics, X (Leipzig, 1991) 43–51, Springer, Berlin (1992)
145.
Ruelle, D.: Dynamical zeta functions for piecewise monotone maps of the interval. CRM Monograph Series, 4, Amer. Math. Soc., Providence, RI (1994) MATH
148.
Ruelle, D.: Dynamical zeta functions and transfer operators. Notices Amer. Math. Soc. 49, 887–895 (2002) MathSciNetMATH
150.
151.
Rugh, H.H.: Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems. Ergodic Theory Dynam. Systems 16, 805–819 (1996) MathSciNetCrossRef
152.
153.
Rugh, H.H.: The Milnor–Thurston determinant and the Ruelle transfer operator. Comm. Math. Phys. 342, 603–614 (2016) MathSciNetCrossRef
160.
Slipantschuk, J., Bandtlow, O.F., Just, W.: Analytic expanding circle maps with explicit spectra. Nonlinearity 26, 3231–3245 (2013) MathSciNetCrossRef
161.
Slipantschuk, J., Bandtlow, O.F., Just, W.: Complete spectral data for analytic Anosov maps of the torus. Nonlinearity 30, 2667–2686 (2017) MathSciNetCrossRef
167.
Thomine, D.: A spectral gap for transfer operators of piecewise expanding maps. Discrete and Continuous Dynamical Systems (A) 30, 917–944 (2011) MathSciNetCrossRef
172.
Tsujii, M.: Decay of correlations in suspension semi-flows of angle multiplying maps. Ergodic Theory Dynam. Systems. 28, 291–317 (2008) MathSciNetCrossRef
175.
Metadaten
Titel
Smooth expanding maps: Dynamical determinants
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_3

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