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February  2012, 32(2): 487-497. doi: 10.3934/dcds.2012.32.487

Pressures for asymptotically sub-additive potentials under a mistake function

Wen-Chiao Cheng 1, , Yun Zhao 2, and  Yongluo Cao 3,

1. 

Department of Applied Mathematics, Chinese Culture University, Yangmingshan, Taipei, 11114, Taiwan
2. 

Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China
3. 

Department of Mathematics, Suzhou University, Suzhou 215006, Jiangsu

Received  September 2010 Revised  November 2010 Published  September 2011

This paper defines the pressure for asymptotically sub-additive potentials under a mistake function, including the measure-theoretical and the topological versions. Using the advanced techniques of ergodic theory and topological dynamics, we reveal a variational principle for the new defined topological pressure without any additional conditions on the potentials and the compact metric space.
Keywords: asymptotically sub-additive., topological pressure, Variational principle.
Mathematics Subject Classification: Primary: 37D35; Secondary: 37A3.
Citation: Wen-Chiao Cheng, Yun Zhao, Yongluo Cao. Pressures for asymptotically sub-additive potentials under a mistake function. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 487-497. doi: 10.3934/dcds.2012.32.487
References:
[1]

L. M. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. Dynam. Syst., 16 (1996), 871-927. doi: 10.1017/S0143385700010117.  Google Scholar

[2]

L. M. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Continuous Dynam. Systems, 16 (2006), 279-305.  Google Scholar

[3]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture notes in Math., 470, Springer-Verlag, Berlin, 1975.  Google Scholar

[4]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Haustes Études Sci. Publ. Math., 50 (1979), 11-25.  Google Scholar

[5]

Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Continuous Dynam. Systems A, 20 (2008), 639-657.  Google Scholar

[6]

K. Falconer, A sub-additive thermodynamic formalism for mixing repellers, J. Phys. A, 21 (1988), L737-L742. doi: 10.1088/0305-4470/21/14/005.  Google Scholar

[7]

D. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Commun. Math. Phys., 297 (2010), 1-43. doi: 10.1007/s00220-010-1031-x.  Google Scholar

[8]

L. He, J. Lv and L. Zhou, Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica, Engl. Ser., 20 (2004), 709-718. doi: 10.1007/s10114-004-0368-5.  Google Scholar

[9]

A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. IHES, 51 (1980), 137-173. Google Scholar

[10]

A. Katok and B. Hasselblatt, "An Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and Its Applications, 54, Cambridge University Press, Cambridge, 1995. Google Scholar

[11]

A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Continuous Dynam. Systems A, 16 (2006), 435-454.  Google Scholar

[12]

Y. Pesin and B. Pitskel', Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen., 18 (1984), 50-63, 96. doi: 10.1007/BF01083692.  Google Scholar

[13]

Y. Pesin, Dimension type characteristics for invariant sets of dynamical systems, Russian Math. Surveys, 43 (1988), 111-151. doi: 10.1070/RM1988v043n04ABEH001892.  Google Scholar

[14]

Y. Pesin, "Dimension Theory in Dynamical Systems, Contemporary Views and Applications," University of Chicago Press, Chicago, 1997. Google Scholar

[15]

C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956. doi: 10.1017/S0143385706000824.  Google Scholar

[16]

C. Pfister and W. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the $\beta$-shifts, Nonlinearity, 18 (2005), 237-261.  Google Scholar

[17]

D. Ruelle, Statistical mechanics on a compact set with $Z^{\upsilon}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251. doi: 10.2307/1996437.  Google Scholar

[18]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley Publishing Co., Reading Mass., 1978. Google Scholar

[19]

D. Thompson, Irregular sets, the $\beta-$transformation and the almost specification property,, preprint, ().   Google Scholar

[20]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[21]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682.  Google Scholar

[22]

G. Zhang, Variational principles of pressure, Discrete Continuous Dynam. Systems A, 24 (2009), 1409-1435.  Google Scholar

[23]

Y. Zhao and Y. Cao, Measure-theoretic pressure for subadditive potentials, Nonlinear Analysis, 70 (2009), 2237-2247. doi: 10.1016/j.na.2008.03.003.  Google Scholar

[24]

Y. Zhao, L. Zhang and Y. Cao, The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials, Nonlinear Analysis, 74 (2011), 5015-5022. Google Scholar

show all references

References:
[1]

L. M. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. Dynam. Syst., 16 (1996), 871-927. doi: 10.1017/S0143385700010117.  Google Scholar

[2]

L. M. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Continuous Dynam. Systems, 16 (2006), 279-305.  Google Scholar

[3]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture notes in Math., 470, Springer-Verlag, Berlin, 1975.  Google Scholar

[4]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Haustes Études Sci. Publ. Math., 50 (1979), 11-25.  Google Scholar

[5]

Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Continuous Dynam. Systems A, 20 (2008), 639-657.  Google Scholar

[6]

K. Falconer, A sub-additive thermodynamic formalism for mixing repellers, J. Phys. A, 21 (1988), L737-L742. doi: 10.1088/0305-4470/21/14/005.  Google Scholar

[7]

D. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Commun. Math. Phys., 297 (2010), 1-43. doi: 10.1007/s00220-010-1031-x.  Google Scholar

[8]

L. He, J. Lv and L. Zhou, Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica, Engl. Ser., 20 (2004), 709-718. doi: 10.1007/s10114-004-0368-5.  Google Scholar

[9]

A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. IHES, 51 (1980), 137-173. Google Scholar

[10]

A. Katok and B. Hasselblatt, "An Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and Its Applications, 54, Cambridge University Press, Cambridge, 1995. Google Scholar

[11]

A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Continuous Dynam. Systems A, 16 (2006), 435-454.  Google Scholar

[12]

Y. Pesin and B. Pitskel', Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen., 18 (1984), 50-63, 96. doi: 10.1007/BF01083692.  Google Scholar

[13]

Y. Pesin, Dimension type characteristics for invariant sets of dynamical systems, Russian Math. Surveys, 43 (1988), 111-151. doi: 10.1070/RM1988v043n04ABEH001892.  Google Scholar

[14]

Y. Pesin, "Dimension Theory in Dynamical Systems, Contemporary Views and Applications," University of Chicago Press, Chicago, 1997. Google Scholar

[15]

C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956. doi: 10.1017/S0143385706000824.  Google Scholar

[16]

C. Pfister and W. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the $\beta$-shifts, Nonlinearity, 18 (2005), 237-261.  Google Scholar

[17]

D. Ruelle, Statistical mechanics on a compact set with $Z^{\upsilon}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251. doi: 10.2307/1996437.  Google Scholar

[18]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley Publishing Co., Reading Mass., 1978. Google Scholar

[19]

D. Thompson, Irregular sets, the $\beta-$transformation and the almost specification property,, preprint, ().   Google Scholar

[20]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[21]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682.  Google Scholar

[22]

G. Zhang, Variational principles of pressure, Discrete Continuous Dynam. Systems A, 24 (2009), 1409-1435.  Google Scholar

[23]

Y. Zhao and Y. Cao, Measure-theoretic pressure for subadditive potentials, Nonlinear Analysis, 70 (2009), 2237-2247. doi: 10.1016/j.na.2008.03.003.  Google Scholar

[24]

Y. Zhao, L. Zhang and Y. Cao, The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials, Nonlinear Analysis, 74 (2011), 5015-5022. Google Scholar

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