American Institute of Mathematical Sciences

Advanced
September  2018, 38(9): 4693-4713. doi: 10.3934/dcds.2018206

Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential

Yunping Jiang 1,2, and  Yuan-Ling Ye 3,,

1. 

Department of Mathematics, Queens College of the City University of New York (CUNY), Flushing, NY 11367-1597, USA
2. 

Department of Mathematics, CUNY Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA
3. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

* Corresponding author

Received  December 2017 Published  June 2018

Fund Project: This material is based upon work supported by the National Science Foundation. Yunping Jiang is partially supported by a collaboration grant from the Simons Foundation (grant number 523341) and PSC-CUNY awards. Both the two authors are partially supported by a grant from NSFC (grant number 11571122)

We study the convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Even without uniformly bounded distortion in this case, which makes the study much harder, we are still able to obtain a very nice estimation of the convergence speed under a certain quasi-gap condition.

Keywords: Non-uniformly expanding dynamical system, weakly contractive IFS, Dini continuity, Ruelle operator (positive transfer operator, Ruelle's Perron-Frobenius operator), convergence speed, decay rate of correlation, quasi-gap condition.
Mathematics Subject Classification: Primary: 37C30; Secondary: 37C40.
Citation: Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206
References:
[1]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633. Google Scholar

[2]

R. Bowen, Equilibrium States And the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math, Vol. 470 (1975), Springer, Berlin. Google Scholar

[3]

A. Fan, A proof of the Ruelle theorem, Reviews Math. Phys., 7 (1995), 1241-1247. doi: 10.1142/S0129055X95000451. Google Scholar

[4]

A. Fan and Y. Jiang, On Ruelle-Perron-Frobenius operators. Ⅰ. Ruelle theorem, Commun. Math. Phys., 223 (2001), 125-141. doi: 10.1007/s002200100538. Google Scholar

[5]

A. Fan and Y. Jiang, On Ruelle-Perron-Frobenius operators Ⅱ. Convergence speeds, Commun. Math. Phys., 223 (2001), 143-159. doi: 10.1007/s002200100539. Google Scholar

[6]

A. Fan and M. Pollicott, Non-homogeneous equilibrium states and convergence speeds of averaging operators, Math. Proc. Cambridge Philos. Soc., 129 (2000), 99-115. doi: 10.1017/S0305004100004485. Google Scholar

[7]

P. Ferrero and B. Schmitt, Ruelle's Perron-Frobenius theorem and projective metrics. Coll. Math. Soc., János Bolyai, 27 (1979), 333–336Google Scholar

[8]

H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergod. Th. & Dynam. Sys., 24 (2004), 495-524. doi: 10.1017/S0143385703000671. Google Scholar

[9]

Y. Jiang, A Proof of existence and simplicity of a maximal eigenvalue for Ruelle-Perron-Frobenius operators, Lett. Math. Phys., 48 (1999), 211-219. doi: 10.1023/A:1007595323704. Google Scholar

[10]

Y. Jiang, Nanjing Lecture Notes In Dynamical Systems. Part One: Transfer Operators in Thermodynamical Formalism. FIM Preprint Series, ETH-Zurich, June 2000.Google Scholar

[11]

Y. Jiang and V. Maume-Deschamps, RPF operators for non-Hölder potentials on an arbitrary metric space. Unpublished Note, 1999.Google Scholar

[12]

Y. Jiang and Y. Ye, Ruelle operator theorem for non-expansive systems, Ergod. Th. and Dynam. Sys., 30 (2010), 469-487. doi: 10.1017/S014338570900025X. Google Scholar

[13]

K. Lau and Y. Ye, Ruelle operator with nonexpansive IFS, Studia Math., 148 (2001), 143-169. doi: 10.4064/sm148-2-4. Google Scholar

[14]

C. Liverani, Decay of correlations, Ann. Math., 142 (1995), 239-301. doi: 10.2307/2118636. Google Scholar

[15]

D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Commun. Math. Phys., 9 (1968), 267-278. doi: 10.1007/BF01654281. Google Scholar

[16]

D. Ruelle, A measure associated with Axiom A attractors, Am. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. Google Scholar

[17]

P. Walters, Ruelle's operator theorem and g-measure, Trans. Amer. Math. Soc., 214 (1975), 375-387. doi: 10.2307/1997113. Google Scholar

[18]

P. Walters, Convergence of the Ruelle operator for a function satisfying Bowen's condition, Trans. Amer. Math. Soc., 353 (2001), 327-347. doi: 10.1090/S0002-9947-00-02656-8. Google Scholar

[19]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982. Google Scholar

[20]

Y. Ye, Ruelle operator with weakly contractive iterated function systems, Ergod. Th. and Dynam. Sys., 33 (2013), 1265-1290. doi: 10.1017/S0143385712000211. Google Scholar

[21]

Y. Ye, Multifractal analysis of non-uniformly contracting iterated function systems, Nonlinearity, 30 (2017), 1708-1733. doi: 10.1088/1361-6544/aa639e. Google Scholar

show all references

References:
[1]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633. Google Scholar

[2]

R. Bowen, Equilibrium States And the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math, Vol. 470 (1975), Springer, Berlin. Google Scholar

[3]

A. Fan, A proof of the Ruelle theorem, Reviews Math. Phys., 7 (1995), 1241-1247. doi: 10.1142/S0129055X95000451. Google Scholar

[4]

A. Fan and Y. Jiang, On Ruelle-Perron-Frobenius operators. Ⅰ. Ruelle theorem, Commun. Math. Phys., 223 (2001), 125-141. doi: 10.1007/s002200100538. Google Scholar

[5]

A. Fan and Y. Jiang, On Ruelle-Perron-Frobenius operators Ⅱ. Convergence speeds, Commun. Math. Phys., 223 (2001), 143-159. doi: 10.1007/s002200100539. Google Scholar

[6]

A. Fan and M. Pollicott, Non-homogeneous equilibrium states and convergence speeds of averaging operators, Math. Proc. Cambridge Philos. Soc., 129 (2000), 99-115. doi: 10.1017/S0305004100004485. Google Scholar

[7]

P. Ferrero and B. Schmitt, Ruelle's Perron-Frobenius theorem and projective metrics. Coll. Math. Soc., János Bolyai, 27 (1979), 333–336Google Scholar

[8]

H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergod. Th. & Dynam. Sys., 24 (2004), 495-524. doi: 10.1017/S0143385703000671. Google Scholar

[9]

Y. Jiang, A Proof of existence and simplicity of a maximal eigenvalue for Ruelle-Perron-Frobenius operators, Lett. Math. Phys., 48 (1999), 211-219. doi: 10.1023/A:1007595323704. Google Scholar

[10]

Y. Jiang, Nanjing Lecture Notes In Dynamical Systems. Part One: Transfer Operators in Thermodynamical Formalism. FIM Preprint Series, ETH-Zurich, June 2000.Google Scholar

[11]

Y. Jiang and V. Maume-Deschamps, RPF operators for non-Hölder potentials on an arbitrary metric space. Unpublished Note, 1999.Google Scholar

[12]

Y. Jiang and Y. Ye, Ruelle operator theorem for non-expansive systems, Ergod. Th. and Dynam. Sys., 30 (2010), 469-487. doi: 10.1017/S014338570900025X. Google Scholar

[13]

K. Lau and Y. Ye, Ruelle operator with nonexpansive IFS, Studia Math., 148 (2001), 143-169. doi: 10.4064/sm148-2-4. Google Scholar

[14]

C. Liverani, Decay of correlations, Ann. Math., 142 (1995), 239-301. doi: 10.2307/2118636. Google Scholar

[15]

D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Commun. Math. Phys., 9 (1968), 267-278. doi: 10.1007/BF01654281. Google Scholar

[16]

D. Ruelle, A measure associated with Axiom A attractors, Am. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. Google Scholar

[17]

P. Walters, Ruelle's operator theorem and g-measure, Trans. Amer. Math. Soc., 214 (1975), 375-387. doi: 10.2307/1997113. Google Scholar

[18]

P. Walters, Convergence of the Ruelle operator for a function satisfying Bowen's condition, Trans. Amer. Math. Soc., 353 (2001), 327-347. doi: 10.1090/S0002-9947-00-02656-8. Google Scholar

[19]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982. Google Scholar

[20]

Y. Ye, Ruelle operator with weakly contractive iterated function systems, Ergod. Th. and Dynam. Sys., 33 (2013), 1265-1290. doi: 10.1017/S0143385712000211. Google Scholar

[21]

Y. Ye, Multifractal analysis of non-uniformly contracting iterated function systems, Nonlinearity, 30 (2017), 1708-1733. doi: 10.1088/1361-6544/aa639e. Google Scholar

[1]

Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003
[2]

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
[3]

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
[4]

Martin Lustig, Caglar Uyanik. Perron-Frobenius theory and frequency convergence for reducible substitutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 355-385. doi: 10.3934/dcds.2017015
[5]

Yuan-Ling Ye. Non-uniformly expanding dynamical systems: Multi-dimension. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2511-2553. doi: 10.3934/dcds.2019106
[6]

Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007
[7]

Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457
[8]

Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007
[9]

Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016
[10]

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
[11]

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
[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]

José F. Alves. Non-uniformly expanding dynamics: Stability from a probabilistic viewpoint. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 363-375. doi: 10.3934/dcds.2001.7.363
[14]

Marianne Akian, Stéphane Gaubert, Antoine Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009
[15]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119
[16]

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
[17]

Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806
[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 (50)
  • HTML views (57)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences