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Regular decay of ball diameters and spectra of Ruelle operators for contact Anosov flows


Author: Luchezar Stoyanov
Journal: Proc. Amer. Math. Soc. 140 (2012), 3463-3478
MSC (2010): Primary 37D20, 37D25
DOI: https://doi.org/10.1090/S0002-9939-2012-11637-5
Published electronically: March 13, 2012
MathSciNet review: 2929015
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Abstract: For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls $ B^s(x,\epsilon )$ on local stable manifolds at Lyapunov regular points $ x$. We prove that this decay rate is similar for all sufficiently small values of $ \epsilon > 0$. From this and the main result in an earlier paper, we derive strong spectral estimates for Ruelle transfer operators for contact Anosov flows with Lipschitz local stable holonomy maps. These apply in particular to geodesic flows on compact locally symmetric manifolds of strictly negative curvature. As is now well known, such spectral estimates have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, asymptotics of closed orbit counting functions, etc.


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Luchezar Stoyanov
Affiliation: School of Mathematics, University of Western Australia, Crawley, WA 6009, Australia
Email: luchezar.stoyanov@uwa.edu.au

DOI: https://doi.org/10.1090/S0002-9939-2012-11637-5
Received by editor(s): April 6, 2011
Published electronically: March 13, 2012
Communicated by: Yingfei Yi
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.