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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, 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), 333–336 Google 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), 333–336 Google 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

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