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

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