Regular decay of ball diameters and spectra of Ruelle operators for contact Anosov flows
Author:
Luchezar Stoyanov
Journal:
Proc. Amer. Math. Soc. 140 (2012), 34633478
MSC (2010):
Primary 37D20, 37D25
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 on local stable manifolds at Lyapunov regular points . We prove that this decay rate is similar for all sufficiently small values of . 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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L. Stoyanov, Nonintegrability of open billiard flows and Dolgopyat type estimates, Ergod. Th. & Dynam. Sys. 32 (2012), 295313.
 [St3]
L. Stoyanov, Pinching conditions, linearization and regularity of Axiom A flows, preprint, 2010 (arXiv: math.DS:1010.1594)
 [An]
 N. Anantharaman, Precise counting results for closed orbits of Anosov flows, Ann. Scient. Éc. Norm. Sup. 33 (2000), 3356. MR 1743718 (2002c:37048)
 [B]
 R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429460. MR 0339281 (49:4041)
 [BP]
 L. Barreira and Ya. Pesin, Lyapunov exponents and smooth ergodic theory, University Lecture Series 23, American Mathematical Society, Providence, RI, 2002. MR 1862379 (2003a:37040)
 [BR]
 R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), 181202. MR 0380889 (52:1786)
 [CL]
 E. Coddington and N. Levinson, Theory of ordinary differential equations, McGrawHill, New York, 1955. MR 0069338 (16:1022b)
 [D]
 D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), 357390. MR 1626749 (99g:58073)
 [Ha]
 B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergod. Th. &Dynam. Sys. 14 (1994), 645666. MR 1304137 (95j:58130)
 [HPS]
 M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Springer Lecture Notes in Mathematics, Vol. 583, 1977. MR 0501173 (58:18595)
 [KH]
 A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge, 1995. MR 1326374 (96c:58055)
 [LY]
 F. Ledrappier and L.S. Young, The metric entropy of diffeomorphisms: Part I: Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2) 122 (1985), 509539; Part II: Relations between entropy, exponents and dimension, Ann. of Math. (2) 122 (1985), 540574. MR 0819556 (87i:58101a); MR 0819557 (87i:58101b)
 [PP]
 W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187188 (1990). MR 1085356 (92f:58141)
 [P1]
 Ya. Pesin, Characteristic exponents and smooth ergodic theory, Russian Mathematical Surveys 32 (1977), 55114.
 [P2]
 Ya. Pesin, Lectures on partial hyperbolicity and stable ergodicity, European Mathematical Society, Zürich, 2004. MR 2068774 (2005j:37039)
 [PeS1]
 V. Petkov and L. Stoyanov, Correlations for pairs of closed trajectories for open billiards, Nonlinearity 22 (2009), 26572679. MR 2550690 (2011a:37080)
 [PeS2]
 V. Petkov and L. Stoyanov, Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function, Analysis and PDE 3 (2010), 427489. MR 2718260 (2012a:35220)
 [PeS3]
 V. Petkov and L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals, Comm. Math. Phys. 310 (2012), 675704.
 [PoS1]
 M. Pollicott and R. Sharp, Exponential error terms for growth functions of negatively curved surfaces, Amer. J. Math. 120 (1998), 10191042. MR 1646052 (99h:58148)
 [PoS2]
 M. Pollicott and R. Sharp, Asymptotic expansions for closed orbits in homology classes, Geom. Dedicata 87 (2001), 123160. MR 1866845 (2003b:37051)
 [PoS3]
 M. Pollicott and R. Sharp, Correlations for pairs of closed geodesics, Invent. Math. 163 (2006), 124. MR 2208416 (2007a:37036)
 [PS]
 C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989), 154. MR 983869 (90h:58057)
 [PSW]
 C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J. 86 (1997), 517546; Correction: Duke Math. J. 105 (2000), 105106. MR 1432307 (97m:58155); MR 1788044 (2001h:37057)
 [Ra]
 M. Ratner, Markov partitions for Anosov flows on dimensional manifolds, Israel J. Math. 15 (1973), 92114. MR 0339282 (49:4042)
 [St1]
 L. Stoyanov, Spectra of Ruelle transfer operators for Axiom A flows, Nonlinearity 24 (2011), 10891120. MR 2776112
 [St2]
 L. Stoyanov, Nonintegrability of open billiard flows and Dolgopyat type estimates, Ergod. Th. & Dynam. Sys. 32 (2012), 295313.
 [St3]
 L. Stoyanov, Pinching conditions, linearization and regularity of Axiom A flows, preprint, 2010 (arXiv: math.DS:1010.1594)
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Additional Information
Luchezar Stoyanov
Affiliation:
School of Mathematics, University of Western Australia, Crawley, WA 6009, Australia
Email:
luchezar.stoyanov@uwa.edu.au
DOI:
http://dx.doi.org/10.1090/S000299392012116375
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.
