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Ruelle transfer operators with two complex parameters and applications
1. | Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France |
2. | School of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009 |
References:
[1] |
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes in Maths. 470, Springer-Verlag, Berlin, 1975. |
[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] |
J. M. Hannay and A. M. Ozorio de Almeida, Periodic orbits and a correlation function for the semiclassical density of states, J. Phys. A, 17 (1984), 3429-3440.
doi: 10.1088/0305-4470/17/18/013. |
[6] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
[7] |
A. Katsuda and T. Sunada, Closed orbits in homology class, Publ. Math. IHES, 71 (1990), 5-32. |
[8] |
S. Lalley, Distribution of period orbits of symbolic and Axiom A flows, Adv. Appl. Math., 8 (1987), 154-193.
doi: 10.1016/0196-8858(87)90012-1. |
[9] |
F. Naud, Expanding maps on Cantor sets and analytic continuation of zeta function, Ann. Sci. Ec. Norm. Sup., 38 (2005), 116-153.
doi: 10.1016/j.ansens.2004.11.002. |
[10] |
W. Parry, Synchronization of canonical measures for hyperbolic attractors, Comm. Math. Phys., 106 (1986), 267-275.
doi: 10.1007/BF01454975. |
[11] |
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188, (1990). |
[12] |
V. Petkov and L. Stoyanov, Sharp large deviations for some hyperbolic systems, Erg. Th. & Dyn. Sys., 35 (1) (2015), 249-273.
doi: 10.1017/etds.2013.48. |
[13] |
V. Petkov and L. Stoyanov, Ruelle operators with two parameters and applications, C. R. Acad. Sci. Paris, Ser. I, 353 (7) (2015), 595-599.
doi: 10.1016/j.crma.2015.04.005. |
[14] |
M. Pollicott, On the rate of mixing of Axiom A flows, Invent. Math., 81 (1985), 413-426.
doi: 10.1007/BF01388579. |
[15] |
M. Pollicott, A note on exponential mixing for Gibbs measures and counting weighted periodic orbits for geodesic flows, Preprint, 2014. |
[16] |
M. Pollicott and R. Sharp, Exponential error terms for growth functions on negatively curved surfaces, Amer. J. Math., 120 (1998), 1019-1042.
doi: 10.1353/ajm.1998.0041. |
[17] |
M. Pollicott and R. Sharp, Large deviations, fluctuations and shrinking intervals, Comm. Math. Phys., 290 (2009), 321-324.
doi: 10.1007/s00220-008-0725-9. |
[18] |
M. Pollicott and R. Sharp, On the Hannay-Ozorio de Almeida sum formula, Dynamics, games and science. II, 575-590, Springer Proc. Math., 2, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-14788-3_41. |
[19] |
D. Ruelle, An extension of the theory of Fredholm determinants, Publ. Math. IHES, 72 (1990), 175-193. |
[20] |
L. Stoyanov, Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows, Amer. J. Math., 123 (2001), 715-759.
doi: 10.1353/ajm.2001.0029. |
[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, 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. |
[23] |
S. Waddington, Large deviations for Anosov flows, Ann. Inst. H. Poincaré, Analyse non-linéaire, 13 (1996), 445-484. |
[24] |
P. Wright, Ruelle's lemma and Ruelle zeta functions, Asymptotic Analysis, 80 (2012), 223-236. |
show all references
References:
[1] |
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes in Maths. 470, Springer-Verlag, Berlin, 1975. |
[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] |
J. M. Hannay and A. M. Ozorio de Almeida, Periodic orbits and a correlation function for the semiclassical density of states, J. Phys. A, 17 (1984), 3429-3440.
doi: 10.1088/0305-4470/17/18/013. |
[6] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
[7] |
A. Katsuda and T. Sunada, Closed orbits in homology class, Publ. Math. IHES, 71 (1990), 5-32. |
[8] |
S. Lalley, Distribution of period orbits of symbolic and Axiom A flows, Adv. Appl. Math., 8 (1987), 154-193.
doi: 10.1016/0196-8858(87)90012-1. |
[9] |
F. Naud, Expanding maps on Cantor sets and analytic continuation of zeta function, Ann. Sci. Ec. Norm. Sup., 38 (2005), 116-153.
doi: 10.1016/j.ansens.2004.11.002. |
[10] |
W. Parry, Synchronization of canonical measures for hyperbolic attractors, Comm. Math. Phys., 106 (1986), 267-275.
doi: 10.1007/BF01454975. |
[11] |
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188, (1990). |
[12] |
V. Petkov and L. Stoyanov, Sharp large deviations for some hyperbolic systems, Erg. Th. & Dyn. Sys., 35 (1) (2015), 249-273.
doi: 10.1017/etds.2013.48. |
[13] |
V. Petkov and L. Stoyanov, Ruelle operators with two parameters and applications, C. R. Acad. Sci. Paris, Ser. I, 353 (7) (2015), 595-599.
doi: 10.1016/j.crma.2015.04.005. |
[14] |
M. Pollicott, On the rate of mixing of Axiom A flows, Invent. Math., 81 (1985), 413-426.
doi: 10.1007/BF01388579. |
[15] |
M. Pollicott, A note on exponential mixing for Gibbs measures and counting weighted periodic orbits for geodesic flows, Preprint, 2014. |
[16] |
M. Pollicott and R. Sharp, Exponential error terms for growth functions on negatively curved surfaces, Amer. J. Math., 120 (1998), 1019-1042.
doi: 10.1353/ajm.1998.0041. |
[17] |
M. Pollicott and R. Sharp, Large deviations, fluctuations and shrinking intervals, Comm. Math. Phys., 290 (2009), 321-324.
doi: 10.1007/s00220-008-0725-9. |
[18] |
M. Pollicott and R. Sharp, On the Hannay-Ozorio de Almeida sum formula, Dynamics, games and science. II, 575-590, Springer Proc. Math., 2, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-14788-3_41. |
[19] |
D. Ruelle, An extension of the theory of Fredholm determinants, Publ. Math. IHES, 72 (1990), 175-193. |
[20] |
L. Stoyanov, Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows, Amer. J. Math., 123 (2001), 715-759.
doi: 10.1353/ajm.2001.0029. |
[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, 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. |
[23] |
S. Waddington, Large deviations for Anosov flows, Ann. Inst. H. Poincaré, Analyse non-linéaire, 13 (1996), 445-484. |
[24] |
P. Wright, Ruelle's lemma and Ruelle zeta functions, Asymptotic Analysis, 80 (2012), 223-236. |
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