# American Institute of Mathematical Sciences

February  2013, 33(2): 391-412. doi: 10.3934/dcds.2013.33.391

## Pinching conditions, linearization and regularity of Axiom A flows

 1 School of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009, Australia

Received  June 2011 Revised  May 2012 Published  September 2012

In this paper we study a certain regularity property of $C^2$ Axiom A flows $\phi_t$ over basic sets $Λ$ related to diameters of balls in Bowen's metric, which we call regular distortion along unstable manifolds. The motivation to investigate the latter comes from the study of spectral properties of Ruelle transfer operators in [21]. We prove that if the bottom of the spectrum of $d\phi_t$ over $E^u_{|Λ}$ is point-wisely pinched and integrable, then the flow has regular distortion along unstable manifolds over $Λ$. In the process, under the same conditions, we show that locally the flow is Lipschitz conjugate to its linearization over the `pinched part' of the unstable tangent bundle.
Citation: Luchezar Stoyanov. Pinching conditions, linearization and regularity of Axiom A flows. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 391-412. doi: 10.3934/dcds.2013.33.391
##### References:
 [1] N. Anantharaman, Precise counting results for closed orbits of Anosov flows, Ann. Scient. Éc. Norm. Sup., 33 (2000), 33-56. [2] R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793. [3] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848. [4] D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math., 147 (1998), 357-390. doi: 10.2307/121012. [5] M. Guysinsky, B. Hasselblatt and V. Rayskin, Differentiability of the Hartman-Grobman linearization, Discr. Cont. Dyn. Syst., 9 (2003), 979-984. [6] B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergod. Th. & Dynam. Sys., 14 (1994), 645-666. [7] B. Hasselblatt, Regularity of the Anosov splitting, Ergod. Th. & Dynam. Sys., 17 (1997), 169-172. doi: 10.1017/S0143385797069757. [8] M. Hirsch and C. Pugh, Smoothness of horocycle foliations, J. Differential Geometry, 10 (1975), 225-238. [9] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Springer Lecture Notes in Mathematics, 583 1977. [10] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Cambridge Univ. Press, Cambridge, 1995. [11] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astéerisque, 187-188 (1990), 268 pp. [12] Ya. Pesin, "Lectures on Partial Hyperbolicity and Stable Ergodicity,'' European Mathematical Society, Zürich, 2004. [13] V. Petkov and L. Stoyanov, Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function, Analysis and PDE, 3 (2010), 427-489. [14] V. Petkov and L. Stoyanov, Correlations for pairs of closed trajectories in open billiards, Nonlinearity, 22 (2009), 2657-2679. doi: 10.1088/0951-7715/22/11/005. [15] V. Petkov and L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals, Commun. Math. Phys., 310 (2012), 675-704. doi: 10.1007/s00220-012-1419-x. [16] M. Pollicott and R. Sharp, Exponential error terms for growth functions of negatively curved surfaces, Amer. J. Math., 120 (1998), 1019-1042. doi: 10.1353/ajm.1998.0041. [17] M. Pollicott and R. Sharp, Asymptotic expansions for closed orbits in homology classes, Geom. Dedicata, 87 (2001), 123-160. doi: 10.1023/A:1012097314447. [18] M. Pollicott and R. Sharp, Correlations for pairs of closed geodesics, Invent. Math., 163 (2006), 1-24. doi: 10.1007/s00222-004-0427-7. [19] C. Pugh and M. Shub, Linearization of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10 (1970), 187-198. doi: 10.1007/BF01403247. [20] C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546; Correction: Duke Math. J., 105 (2000), 105-106. doi: 10.1215/S0012-7094-97-08616-6. [21] L. Stoyanov, Spectra of Ruelle transfer operators for Axiom A flows, Nonlinearity, 24 (2011), 1089-1120. doi: 10.1088/0951-7715/24/4/005. [22] L. Stoyanov, Non-integrability of open billiard flows and Dolgopyat type estimates, Ergod. Th. & Dynam. Sys., 32 (2012), 295-313. doi: 10.1017/S0143385710000933.

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##### References:
 [1] N. Anantharaman, Precise counting results for closed orbits of Anosov flows, Ann. Scient. Éc. Norm. Sup., 33 (2000), 33-56. [2] R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793. [3] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848. [4] D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math., 147 (1998), 357-390. doi: 10.2307/121012. [5] M. Guysinsky, B. Hasselblatt and V. Rayskin, Differentiability of the Hartman-Grobman linearization, Discr. Cont. Dyn. Syst., 9 (2003), 979-984. [6] B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergod. Th. & Dynam. Sys., 14 (1994), 645-666. [7] B. Hasselblatt, Regularity of the Anosov splitting, Ergod. Th. & Dynam. Sys., 17 (1997), 169-172. doi: 10.1017/S0143385797069757. [8] M. Hirsch and C. Pugh, Smoothness of horocycle foliations, J. Differential Geometry, 10 (1975), 225-238. [9] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Springer Lecture Notes in Mathematics, 583 1977. [10] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Cambridge Univ. Press, Cambridge, 1995. [11] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astéerisque, 187-188 (1990), 268 pp. [12] Ya. Pesin, "Lectures on Partial Hyperbolicity and Stable Ergodicity,'' European Mathematical Society, Zürich, 2004. [13] V. Petkov and L. Stoyanov, Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function, Analysis and PDE, 3 (2010), 427-489. [14] V. Petkov and L. Stoyanov, Correlations for pairs of closed trajectories in open billiards, Nonlinearity, 22 (2009), 2657-2679. doi: 10.1088/0951-7715/22/11/005. [15] V. Petkov and L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals, Commun. Math. Phys., 310 (2012), 675-704. doi: 10.1007/s00220-012-1419-x. [16] M. Pollicott and R. Sharp, Exponential error terms for growth functions of negatively curved surfaces, Amer. J. Math., 120 (1998), 1019-1042. doi: 10.1353/ajm.1998.0041. [17] M. Pollicott and R. Sharp, Asymptotic expansions for closed orbits in homology classes, Geom. Dedicata, 87 (2001), 123-160. doi: 10.1023/A:1012097314447. [18] M. Pollicott and R. Sharp, Correlations for pairs of closed geodesics, Invent. Math., 163 (2006), 1-24. doi: 10.1007/s00222-004-0427-7. [19] C. Pugh and M. Shub, Linearization of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10 (1970), 187-198. doi: 10.1007/BF01403247. [20] C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546; Correction: Duke Math. J., 105 (2000), 105-106. doi: 10.1215/S0012-7094-97-08616-6. [21] L. Stoyanov, Spectra of Ruelle transfer operators for Axiom A flows, Nonlinearity, 24 (2011), 1089-1120. doi: 10.1088/0951-7715/24/4/005. [22] L. Stoyanov, Non-integrability of open billiard flows and Dolgopyat type estimates, Ergod. Th. & Dynam. Sys., 32 (2012), 295-313. doi: 10.1017/S0143385710000933.
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