Ruelle transfer operators with two complex parameters and applications

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November 2016, 36(11): 6413-6451. doi: 10.3934/dcds.2016077

Ruelle transfer operators with two complex parameters and applications

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

Received October 2015 Revised June 2016 Published August 2016

Abstract
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For a $C^2$ Axiom A flow $\phi_t: M \longrightarrow M$ on a Riemannian manifold $M$ and a basic set $\Lambda$ for $\phi_t$ we consider the Ruelle transfer operator $L_{f - s \tau + z g}$, where $f$ and $g$ are real-valued Hölder functions on $\Lambda$, $\tau$ is the roof function and $s, z \in \mathbb{C}$ are complex parameters. Under some assumptions about $\phi_t$ we establish estimates for the iterations of this Ruelle operator in the spirit of the estimates for operators with one complex parameter (see [4], [21], [22]). Two cases are covered: (i) for arbitrary Hölder $f,g$ when $|Im z| \leq B |Im s|^\mu$ for some constants $B > 0$, $0 < \mu < 1$ ($\mu = 1$ for Lipschitz $f,g$), (ii) for Lipschitz $f,g$ when $|Im s| \leq B_1 |Im z|$ for some constant $B_1 > 0$ . Applying these estimates, we obtain a non zero analytic extension of the zeta function $\zeta(s, z)$ for $P_f - \epsilon < Re (s) < P_f$ and $|z|$ small enough with a simple pole at $s = s(z)$. Two other applications are considered as well: the first concerns the Hannay-Ozorio de Almeida sum formula, while the second deals with the asymptotic of the counting function $\pi_F(T)$ for weighted primitive periods of the flow $\phi_t.$

Keywords: Axiom A flow, periods counting function., Ruelle transfer operator, zeta function, basic set.

Mathematics Subject Classification: Primary: 37C30, Secondary: 37D20, 37C3.

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

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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