American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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.  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, Decay of correlations in Anosov flows, Ann. Math., 147 (1998), 357-390.  doi: 10.2307/121012.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar
[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327
[2]

Frédéric Naud. The Ruelle spectrum of generic transfer operators. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2521-2531. doi: 10.3934/dcds.2012.32.2521
[3]

Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729
[4]

Patricia Domínguez, Peter Makienko, Guillermo Sienra. Ruelle operator and transcendental entire maps. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 773-789. doi: 10.3934/dcds.2005.12.773
[5]

Vesselin Petkov, Luchezar Stoyanov. Ruelle transfer operators with two complex parameters and applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6413-6451. doi: 10.3934/dcds.2016077
[6]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[7]

Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881
[8]

Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453
[9]

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805
[10]

Yoshikazu Giga, Hiroyoshi Mitake, Hung V. Tran. Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019228
[11]

Leandro Cioletti, Artur O. Lopes. Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6139-6152. doi: 10.3934/dcds.2017264
[12]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665
[13]

Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155
[14]

Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic & Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245
[15]

Alexander Veretennikov. On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 523-549. doi: 10.3934/dcdsb.2013.18.523
[16]

Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965
[17]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216
[18]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[19]

Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48
[20]

Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (36)
  • HTML views (75)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences