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November  2019, 39(11): 6391-6417. doi: 10.3934/dcds.2019277

Spectral estimates for Ruelle operators with two parameters and sharp large deviations

Vesselin Petkov 1, and  Luchezar Stoyanov 2,

1. 

Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France
2. 

University of Western Australia, Department of Mathematics and Statistics, 35 Stirling Highway, Perth WA 6009, Australia

Received  November 2018 Revised  April 2019 Published  August 2019

We obtain spectral estimates for the iterations of Ruelle operators $ L_{f + (a + {\bf i} b)\tau + (c + {\bf i} d) g} $ with two complex parameters and Hölder continuous functions $ f,\: g $ generalizing the case $ {\rm{Pr}}(f) = 0 $ studied in [9]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [8].

Keywords: Ruelle transfer operator, Axiom A flow, basic set, large deviations.
Mathematics Subject Classification: Primary: 37C30; Secondary: 37D20, 37C35.
Citation: Vesselin Petkov, Luchezar Stoyanov. Spectral estimates for Ruelle operators with two parameters and sharp large deviations. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6391-6417. doi: 10.3934/dcds.2019277
References:
[1]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect. Notes in Maths. 470. Springer-Verlag, Berlin, 2008.

[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, Decay of correlations in Anosov flows, Ann. Math., 147 (1998), 357-390.  doi: 10.2307/121012.

[5] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54. Cambridge Univ. Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.
[6]

S. P. Lalley, Distribution of periodic orbits of symbolic and Axiom A flows, Adv. Appl. Math., 8 (1987), 154-193.  doi: 10.1016/0196-8858(87)90012-1.

[7]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990), 268 pp.

[8]

V. Petkov and L. Stoyanov, Sharp large deviations for some hyperbolic systems, Erg. Th. & Dyn. Sys., 35 (2015), 249-273. doi: 10.1017/etds.2013.48.

[9]

V. Petkov and L. Stoyanov, Ruelle transfer operators with two complex parameters and applications, Discr. Cont. Dyn. Sys. A, 36 (2016), 6413-6451. doi: 10.3934/dcds.2016077.

[10]

M. Pollicott and R. Sharp, Large deviations, fluctuations and shrinking intervals, Comm. Math. Phys., 290 (2009), 321-334.  doi: 10.1007/s00220-008-0725-9.

[11]

L. Stoyanov, Spectra of Ruelle transfer operators for Axiom A flows, Nonlinearity, 24 (2011), 1089-1120.  doi: 10.1088/0951-7715/24/4/005.

[12]

L. Stoyanov, Pinching conditions, linearization and regularity of Axiom A flows, Discr. Cont. Dyn. Sys. A, 33 (2013), 391-412.  doi: 10.3934/dcds.2013.33.391.

[13]

S. Waddington, Large deviations for Anosov flows, Ann. Inst. H. Poincaré, Analyse non-linéaire, 13 (1996), 445-484.  doi: 10.1016/S0294-1449(16)30110-X.

show all references

References:
[1]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect. Notes in Maths. 470. Springer-Verlag, Berlin, 2008.

[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, Decay of correlations in Anosov flows, Ann. Math., 147 (1998), 357-390.  doi: 10.2307/121012.

[5] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54. Cambridge Univ. Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.
[6]

S. P. Lalley, Distribution of periodic orbits of symbolic and Axiom A flows, Adv. Appl. Math., 8 (1987), 154-193.  doi: 10.1016/0196-8858(87)90012-1.

[7]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990), 268 pp.

[8]

V. Petkov and L. Stoyanov, Sharp large deviations for some hyperbolic systems, Erg. Th. & Dyn. Sys., 35 (2015), 249-273. doi: 10.1017/etds.2013.48.

[9]

V. Petkov and L. Stoyanov, Ruelle transfer operators with two complex parameters and applications, Discr. Cont. Dyn. Sys. A, 36 (2016), 6413-6451. doi: 10.3934/dcds.2016077.

[10]

M. Pollicott and R. Sharp, Large deviations, fluctuations and shrinking intervals, Comm. Math. Phys., 290 (2009), 321-334.  doi: 10.1007/s00220-008-0725-9.

[11]

L. Stoyanov, Spectra of Ruelle transfer operators for Axiom A flows, Nonlinearity, 24 (2011), 1089-1120.  doi: 10.1088/0951-7715/24/4/005.

[12]

L. Stoyanov, Pinching conditions, linearization and regularity of Axiom A flows, Discr. Cont. Dyn. Sys. A, 33 (2013), 391-412.  doi: 10.3934/dcds.2013.33.391.

[13]

S. Waddington, Large deviations for Anosov flows, Ann. Inst. H. Poincaré, Analyse non-linéaire, 13 (1996), 445-484.  doi: 10.1016/S0294-1449(16)30110-X.

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