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On Gibbs Measures and Spectra of Ruelle Transfer Operators

Published online by Cambridge University Press:  20 November 2018

Luchezar Stoyanov*
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia. e-mail: luchezar.stoyanov@uwa.edu.au
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Abstract

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We prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Baladi, V., Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, 16. World Scientific, River Edge, NJ, 2000. http://dx.doi.org/10.1142/9789812813633 Google Scholar
[2] Bowen, R., Equilibrium states and the ergodic theory ofAnosov diffeomorphisms. Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 1975. Google Scholar
[3] Dolgopyat, D., On decay of correlations in Anosov flows. Ann. of Math. 147(1998), no. 2, 357390. http://dx.doi.org/10.2307/1 21012 Google Scholar
[4] Naud, E, Dynamics on Cantor sets and analytic properties of zeta functions. Ph.D. thesis, University of Bordeaux I, 2003. Google Scholar
[5] Parry, W. and Pollicott, M., Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187188,1990. Google Scholar
[6] Pollicott, M., A complex Ruelle operator theorem and two counterexamples. Ergodic Theory Dynam. Systems 4(1984), 135146. http://dx.doi.org/10.1017/S0143385700002327 Google Scholar
[7] Ruelle, D., Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9(1968), 267278. http://dx.doi.org/10.1007/BF01654281 Google Scholar
[8] Ruelle, D., A measure associated with Axiom A attractors. Amer. J. Math. 98(1976), 619654. http://dx.doi.org/10.2307/2373810 Google Scholar
[9] Stoyanov, L., On the Ruelle-Perron-Frobenius Theorem. Asymptot Anal. 43(2005), 131150.Google Scholar
[10] Stoyanov, L., Scattering resonances for several small convex bodies and the Lax-Phillips conjecture. Mem. Amer. Math. Soc. 199(2009).Google Scholar
[11] Stoyanov, L., Spectra of Ruelle transfer operators for Axiom A flows. Nonlinearity 24(2011), 10891120. http://dx.doi.org/! 0.1088/0951-771 5/24/4/005 Google Scholar