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Study of a degenerate reaction-diffusion system arising in particle dynamics with aggregation effects
Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential
| 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 |
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.
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. |
| [2] |
R. Bowen,
Equilibrium States And the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math, Vol. 470 (1975), Springer, Berlin. |
| [3] |
A. Fan,
A proof of the Ruelle theorem, Reviews Math. Phys., 7 (1995), 1241-1247.
doi: 10.1142/S0129055X95000451. |
| [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. |
| [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. |
| [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. |
| [7] |
P. Ferrero and B. Schmitt, Ruelle's Perron-Frobenius theorem and projective metrics. Coll. Math. Soc., János Bolyai, 27 (1979), 333–336 |
| [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. |
| [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. |
| [10] |
Y. Jiang, Nanjing Lecture Notes In Dynamical Systems. Part One: Transfer Operators in Thermodynamical Formalism. FIM Preprint Series, ETH-Zurich, June 2000. |
| [11] |
Y. Jiang and V. Maume-Deschamps, RPF operators for non-Hölder potentials on an arbitrary metric space. Unpublished Note, 1999. |
| [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. |
| [13] |
K. Lau and Y. Ye,
Ruelle operator with nonexpansive IFS, Studia Math., 148 (2001), 143-169.
doi: 10.4064/sm148-2-4. |
| [14] |
C. Liverani,
Decay of correlations, Ann. Math., 142 (1995), 239-301.
doi: 10.2307/2118636. |
| [15] |
D. Ruelle,
Statistical mechanics of a one-dimensional lattice gas, Commun. Math. Phys., 9 (1968), 267-278.
doi: 10.1007/BF01654281. |
| [16] |
D. Ruelle,
A measure associated with Axiom A attractors, Am. J. Math., 98 (1976), 619-654.
doi: 10.2307/2373810. |
| [17] |
P. Walters,
Ruelle's operator theorem and g-measure, Trans. Amer. Math. Soc., 214 (1975), 375-387.
doi: 10.2307/1997113. |
| [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. |
| [19] |
P. Walters,
An Introduction to Ergodic Theory, Springer-Verlag, 1982. |
| [20] |
Y. Ye,
Ruelle operator with weakly contractive iterated function systems, Ergod. Th. and Dynam. Sys., 33 (2013), 1265-1290.
doi: 10.1017/S0143385712000211. |
| [21] |
Y. Ye,
Multifractal analysis of non-uniformly contracting iterated function systems, Nonlinearity, 30 (2017), 1708-1733.
doi: 10.1088/1361-6544/aa639e. |
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. |
| [2] |
R. Bowen,
Equilibrium States And the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math, Vol. 470 (1975), Springer, Berlin. |
| [3] |
A. Fan,
A proof of the Ruelle theorem, Reviews Math. Phys., 7 (1995), 1241-1247.
doi: 10.1142/S0129055X95000451. |
| [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. |
| [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. |
| [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. |
| [7] |
P. Ferrero and B. Schmitt, Ruelle's Perron-Frobenius theorem and projective metrics. Coll. Math. Soc., János Bolyai, 27 (1979), 333–336 |
| [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. |
| [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. |
| [10] |
Y. Jiang, Nanjing Lecture Notes In Dynamical Systems. Part One: Transfer Operators in Thermodynamical Formalism. FIM Preprint Series, ETH-Zurich, June 2000. |
| [11] |
Y. Jiang and V. Maume-Deschamps, RPF operators for non-Hölder potentials on an arbitrary metric space. Unpublished Note, 1999. |
| [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. |
| [13] |
K. Lau and Y. Ye,
Ruelle operator with nonexpansive IFS, Studia Math., 148 (2001), 143-169.
doi: 10.4064/sm148-2-4. |
| [14] |
C. Liverani,
Decay of correlations, Ann. Math., 142 (1995), 239-301.
doi: 10.2307/2118636. |
| [15] |
D. Ruelle,
Statistical mechanics of a one-dimensional lattice gas, Commun. Math. Phys., 9 (1968), 267-278.
doi: 10.1007/BF01654281. |
| [16] |
D. Ruelle,
A measure associated with Axiom A attractors, Am. J. Math., 98 (1976), 619-654.
doi: 10.2307/2373810. |
| [17] |
P. Walters,
Ruelle's operator theorem and g-measure, Trans. Amer. Math. Soc., 214 (1975), 375-387.
doi: 10.2307/1997113. |
| [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. |
| [19] |
P. Walters,
An Introduction to Ergodic Theory, Springer-Verlag, 1982. |
| [20] |
Y. Ye,
Ruelle operator with weakly contractive iterated function systems, Ergod. Th. and Dynam. Sys., 33 (2013), 1265-1290.
doi: 10.1017/S0143385712000211. |
| [21] |
Y. Ye,
Multifractal analysis of non-uniformly contracting iterated function systems, Nonlinearity, 30 (2017), 1708-1733.
doi: 10.1088/1361-6544/aa639e. |
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