# American Institute of Mathematical Sciences

October  2013, 33(10): 4401-4410. doi: 10.3934/dcds.2013.33.4401

## On the large deviation rates of non-entropy-approachable measures

 1 Department of Mathematics, Kyoto University, 606-8502, Kyoto, Japan 2 School of Information Environment, Tokyo Denki University, 2-1200 Buseigakuendai, Inzai-shi, Chiba 270-1382

Received  October 2010 Revised  March 2013 Published  April 2013

We construct a non-ergodic maximal entropy measure of a $C^{\infty}$ diffeomorphism with a positive entropy such that neither the entropy nor the large deviation rate of the measure is influenced by that of ergodic measures near it.
Citation: Masayuki Asaoka, Kenichiro Yamamoto. On the large deviation rates of non-entropy-approachable measures. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4401-4410. doi: 10.3934/dcds.2013.33.4401
##### References:
 [1] L. Barreira and Ya. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", University Lecture Series, 23 (2002).   Google Scholar [2] A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", $2^{nd}$ edition, 38 (1998).   Google Scholar [3] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces,", Lecture Notes in Mathematics, 527 (1976).   Google Scholar [4] A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $\mathbbZ^d$-actions,, Comm. Math. Phys., 164 (1994), 433.  doi: 10.1007/BF02101485.  Google Scholar [5] H. Follmer and S. Orey, Large deviations for the empirical field of a Gibbs measure,, Ann. Probab., 16 (1988), 961.  doi: 10.1214/aop/1176991671.  Google Scholar [6] F. Hofbauer, Generic properties of invariant measures for continuous piecewise monotonic transformations,, Monatsh. Math., 106 (1988), 301.  doi: 10.1007/BF01295288.  Google Scholar [7] C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the $\beta$-shifts,, Nonlinearity, 18 (2005), 237.  doi: 10.1088/0951-7715/18/1/013.  Google Scholar [8] Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems,, Comm. Math. Phys., 74 (1980), 189.  doi: 10.1007/BF01197757.  Google Scholar [9] M. Qian, J.-S. Xie and S. Zhu, "Smooth Ergodic Theory for Endomorphisms,", Lecture Notes in Mathematics, 1978 (2009).  doi: 10.1007/978-3-642-01954-8.  Google Scholar [10] K. Sigmund, Generic properties of invariant measures for axiom-A diffeomorphisms,, Invent. Math., 11 (1970), 99.  doi: 10.1007/BF01404606.  Google Scholar [11] K. Yamamoto, On the weaker forms of the specification property and their applications,, Proc. Amer. Math. Soc., 137 (2009), 3807.  doi: 10.1090/S0002-9939-09-09937-7.  Google Scholar [12] L.-S. Young, Some large deviation results for dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525.  doi: 10.2307/2001318.  Google Scholar

show all references

##### References:
 [1] L. Barreira and Ya. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", University Lecture Series, 23 (2002).   Google Scholar [2] A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", $2^{nd}$ edition, 38 (1998).   Google Scholar [3] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces,", Lecture Notes in Mathematics, 527 (1976).   Google Scholar [4] A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $\mathbbZ^d$-actions,, Comm. Math. Phys., 164 (1994), 433.  doi: 10.1007/BF02101485.  Google Scholar [5] H. Follmer and S. Orey, Large deviations for the empirical field of a Gibbs measure,, Ann. Probab., 16 (1988), 961.  doi: 10.1214/aop/1176991671.  Google Scholar [6] F. Hofbauer, Generic properties of invariant measures for continuous piecewise monotonic transformations,, Monatsh. Math., 106 (1988), 301.  doi: 10.1007/BF01295288.  Google Scholar [7] C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the $\beta$-shifts,, Nonlinearity, 18 (2005), 237.  doi: 10.1088/0951-7715/18/1/013.  Google Scholar [8] Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems,, Comm. Math. Phys., 74 (1980), 189.  doi: 10.1007/BF01197757.  Google Scholar [9] M. Qian, J.-S. Xie and S. Zhu, "Smooth Ergodic Theory for Endomorphisms,", Lecture Notes in Mathematics, 1978 (2009).  doi: 10.1007/978-3-642-01954-8.  Google Scholar [10] K. Sigmund, Generic properties of invariant measures for axiom-A diffeomorphisms,, Invent. Math., 11 (1970), 99.  doi: 10.1007/BF01404606.  Google Scholar [11] K. Yamamoto, On the weaker forms of the specification property and their applications,, Proc. Amer. Math. Soc., 137 (2009), 3807.  doi: 10.1090/S0002-9939-09-09937-7.  Google Scholar [12] L.-S. Young, Some large deviation results for dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525.  doi: 10.2307/2001318.  Google Scholar
 [1] Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673 [2] Andreas Strömbergsson. On the deviation of ergodic averages for horocycle flows. Journal of Modern Dynamics, 2013, 7 (2) : 291-328. doi: 10.3934/jmd.2013.7.291 [3] John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16. [4] Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383 [5] Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221 [6] François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275 [7] Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192 [8] Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048 [9] Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289 [10] Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for non-conformal repellers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5763-5780. doi: 10.3934/dcds.2017250 [11] Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2767-2789. doi: 10.3934/dcds.2020149 [12] Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1 [13] Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159 [14] Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61 [15] Peng Zhang. Multiperiod mean semi-absolute deviation interval portfolio selection with entropy constraints. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1169-1187. doi: 10.3934/jimo.2016067 [16] Jean-Baptiste Bardet, Bastien Fernandez. Extensive escape rate in lattices of weakly coupled expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 669-684. doi: 10.3934/dcds.2011.31.669 [17] Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [18] Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215 [19] Manfred Einsiedler, Elon Lindenstrauss. Symmetry of entropy in higher rank diagonalizable actions and measure classification. Journal of Modern Dynamics, 2018, 13: 163-185. doi: 10.3934/jmd.2018016 [20] Yoora Kim, Gang Uk Hwang, Hea Sook Park. Feedback limited opportunistic scheduling and admission control for ergodic rate guarantees over Nakagami-$m$ fading channels. Journal of Industrial & Management Optimization, 2009, 5 (3) : 553-567. doi: 10.3934/jimo.2009.5.553

2018 Impact Factor: 1.143