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

Pinching conditions, linearization and regularity of Axiom A flows

Luchezar Stoyanov 1,

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.
Keywords: pinching, Hyperbolic flow, linearization, regular distortion., basic set.
Mathematics Subject Classification: Primary: 37D20; Secondary: 37A2.
Citation: Luchezar Stoyanov. Pinching conditions, linearization and regularity of Axiom A flows. Discrete & 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.  Google Scholar

[2]

R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793.  Google Scholar

[3]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848.  Google Scholar

[4]

D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math., 147 (1998), 357-390. doi: 10.2307/121012.  Google Scholar

[5]

M. Guysinsky, B. Hasselblatt and V. Rayskin, Differentiability of the Hartman-Grobman linearization, Discr. Cont. Dyn. Syst., 9 (2003), 979-984.  Google Scholar

[6]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergod. Th. & Dynam. Sys., 14 (1994), 645-666.  Google Scholar

[7]

B. Hasselblatt, Regularity of the Anosov splitting, Ergod. Th. & Dynam. Sys., 17 (1997), 169-172. doi: 10.1017/S0143385797069757.  Google Scholar

[8]

M. Hirsch and C. Pugh, Smoothness of horocycle foliations, J. Differential Geometry, 10 (1975), 225-238.  Google Scholar

[9]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Springer Lecture Notes in Mathematics, 583 1977.  Google Scholar

[10]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Cambridge Univ. Press, Cambridge, 1995.  Google Scholar

[11]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astéerisque, 187-188 (1990), 268 pp.  Google Scholar

[12]

Ya. Pesin, "Lectures on Partial Hyperbolicity and Stable Ergodicity,'' European Mathematical Society, Zürich, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

C. Pugh and M. Shub, Linearization of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10 (1970), 187-198. doi: 10.1007/BF01403247.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

N. Anantharaman, Precise counting results for closed orbits of Anosov flows, Ann. Scient. Éc. Norm. Sup., 33 (2000), 33-56.  Google Scholar

[2]

R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793.  Google Scholar

[3]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848.  Google Scholar

[4]

D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math., 147 (1998), 357-390. doi: 10.2307/121012.  Google Scholar

[5]

M. Guysinsky, B. Hasselblatt and V. Rayskin, Differentiability of the Hartman-Grobman linearization, Discr. Cont. Dyn. Syst., 9 (2003), 979-984.  Google Scholar

[6]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergod. Th. & Dynam. Sys., 14 (1994), 645-666.  Google Scholar

[7]

B. Hasselblatt, Regularity of the Anosov splitting, Ergod. Th. & Dynam. Sys., 17 (1997), 169-172. doi: 10.1017/S0143385797069757.  Google Scholar

[8]

M. Hirsch and C. Pugh, Smoothness of horocycle foliations, J. Differential Geometry, 10 (1975), 225-238.  Google Scholar

[9]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Springer Lecture Notes in Mathematics, 583 1977.  Google Scholar

[10]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Cambridge Univ. Press, Cambridge, 1995.  Google Scholar

[11]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astéerisque, 187-188 (1990), 268 pp.  Google Scholar

[12]

Ya. Pesin, "Lectures on Partial Hyperbolicity and Stable Ergodicity,'' European Mathematical Society, Zürich, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

C. Pugh and M. Shub, Linearization of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10 (1970), 187-198. doi: 10.1007/BF01403247.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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