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A note on operator semigroups associated to chaotic flows

Published online by Cambridge University Press:  11 February 2015

OLIVER BUTTERLEY
OLIVER BUTTERLEY*
Affiliation:
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria email [email protected]
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Abstract

The transfer operator associated to a flow (continuous time dynamical system) is a one-parameter operator semigroup. We consider the operator-valued Laplace transform of this one-parameter semigroup. Estimates on the Laplace transform have been used in various settings in order to show the rate at which the flow mixes. Here we consider the case of exponential mixing and the case of rapid mixing (superpolynomial). We develop the operator theory framework amenable to this setting and show that the same estimates may be used to produce results, in terms of the operators, which go beyond the results for the rate of mixing.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 5 , August 2016 , pp. 1396 - 1408
Copyright
© Cambridge University Press, 2015 

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References

Arendt, W., Batty, C., Hieber, M. and Neubrander, F.. Vector-valued Laplace Transforms and Cauchy Problems (Monographs in Mathematics) , 2nd edn. Birkhäuser, Basel, 2011.Google Scholar
Avila, A., Gouëzel, S. and Yoccoz, J.. Exponential mixing for the Teichmüller flow. Publ. Math. Inst. Hautes Études Sci. 104(1) (2006), 143–211.Google Scholar
Baladi, V. and Liverani, C.. Exponential decay of correlations for piecewise cone hyperbolic contact flows. Comm. Math. Phys. 314(3) (2012), 689–773.Google Scholar
Butterley, O.. Expanding semiflows on branched surfaces and one-parameter semigroups of operators. Nonlinearity 25(12) (2012), 3487–3503.Google Scholar
Butterley, O.. Area expanding C1+𝛼 suspension semiflows. Comm. Math. Phys. 325(2) (2014), 803–820.Google Scholar
Butterley, O. and Liverani, C.. Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1(2) (2007), 301–322.CrossRefGoogle Scholar
Butterley, O. and Liverani, C.. Robustly invariant sets in fiber contracting bundle flows. J. Mod. Dyn. 7(2) (2013), 255–267.Google Scholar
Davies, E. B.. Linear Operators and Their Spectra (Cambridge studies in Advanced Mathematics, 106) . Cambridge University Press, Cambridge, 2007.Google Scholar
Demers, M. F. and Zhang, H.-K.. Spectral analysis of the transfer operator for the Lorentz gas. J. Mod. Dyn. 5(4) (2011), 665–709.Google Scholar
Dolgopyat, D.. On decay of correlations in Anosov flows. Ann. of Math. (2) 147 (1998), 357–390.Google Scholar
Dolgopyat, D.. Prevalence of rapid mixing in hyperbolic flows. Ergod. Th. & Dynam. Sys. 18(5) (1998), 1097–1114.CrossRefGoogle Scholar
Dolgopyat, D.. Prevalence of rapid mixing—II: Topological prevalence. Ergod. Th. & Dynam. Sys. 20(4) (2000), 1045–1059.Google Scholar
Field, M., Melbourne, I. and Török, A.. Stability of mixing and rapid mixing for hyperbolic flows. Ann. of Math. (2) 166(1) (2007), 269–291.CrossRefGoogle Scholar
Giulietti, P., Liverani, C. and Pollicott, M.. Anosov flows and dynamical zeta functions. Ann. of Math. (2) 178(2) (2013), 687–773.Google Scholar
Hennion, H.. Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118(2) (1993), 627–634.Google Scholar
Keller, G.. Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems. Trans. Amer. Math. Soc. 314(2) (1989), 433–497.CrossRefGoogle Scholar
Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28(1) (1999), 141–152.Google Scholar
Keller, G. and Liverani, C.. A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps. Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems (Lecture Notes in Physics, 671) . Springer, Berlin, 2005, pp. 115–151.Google Scholar
Liverani, C.. On contact Anosov flows. Ann. of Math. (2) 159 (2004), 1275–1312.Google Scholar
Melbourne, I.. Rapid decay of correlations for nonuniformly hyperbolic flows. Trans. Amer. Math. Soc. 359(5) (2007), 2421–2441.Google Scholar
Melbourne, I.. Decay of correlations for slowly mixing flows. Proc. Lond. Math. Soc. (3) 98(1) (2009), 163–190.Google Scholar
Nussbaum, R. D.. The radius of essential spectrum. Duke Math. J. 37 (1970), 473–478.CrossRefGoogle Scholar
Pollicott, M.. On the rate of mixing of Axiom A flows. Invent. Math. 81(3) (1985), 413–426.Google Scholar
Tsujii, M.. Decay of correlations in suspension semi-flows of angle multiplying maps. Ergod. Th. & Dynam. Sys. 28(1) (2008), 291–317.Google Scholar
Tsujii, M.. Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity 23(7) (2010), 1495–1545.CrossRefGoogle Scholar
Tsujii, M.. Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform. Ergod. Th. & Dynam. Sys. 32(6) (2012), 2083–2118.Google Scholar