Pressures for asymptotically sub-additive potentials under a mistake function

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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

Abstract
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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 and 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.
[2]	L. M. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Continuous Dynam. Systems, 16 (2006), 279-305.
[3]	R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture notes in Math., 470, Springer-Verlag, Berlin, 1975.
[4]	R. Bowen, Hausdorff dimension of quasicircles, Inst. Haustes Études Sci. Publ. Math., 50 (1979), 11-25.
[5]	Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Continuous Dynam. Systems A, 20 (2008), 639-657.
[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.
[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.
[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.
[9]	A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. IHES, 51 (1980), 137-173.
[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.
[11]	A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Continuous Dynam. Systems A, 16 (2006), 435-454.
[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.
[13]	Y. Pesin, Dimension type characteristics for invariant sets of dynamical systems, Russian Math. Surveys, 43 (1988), 111-151. doi: 10.1070/RM1988v043n04ABEH001892.
[14]	Y. Pesin, "Dimension Theory in Dynamical Systems, Contemporary Views and Applications," University of Chicago Press, Chicago, 1997.
[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.
[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.
[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.
[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.
[19]	D. Thompson, Irregular sets, the $\beta-$transformation and the almost specification property, preprint, arXiv:0905.0739v1.
[20]	P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
[21]	P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682.
[22]	G. Zhang, Variational principles of pressure, Discrete Continuous Dynam. Systems A, 24 (2009), 1409-1435.
[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.
[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.

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.
[2]	L. M. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Continuous Dynam. Systems, 16 (2006), 279-305.
[3]	R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture notes in Math., 470, Springer-Verlag, Berlin, 1975.
[4]	R. Bowen, Hausdorff dimension of quasicircles, Inst. Haustes Études Sci. Publ. Math., 50 (1979), 11-25.
[5]	Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Continuous Dynam. Systems A, 20 (2008), 639-657.
[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.
[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.
[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.
[9]	A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. IHES, 51 (1980), 137-173.
[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.
[11]	A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Continuous Dynam. Systems A, 16 (2006), 435-454.
[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.
[13]	Y. Pesin, Dimension type characteristics for invariant sets of dynamical systems, Russian Math. Surveys, 43 (1988), 111-151. doi: 10.1070/RM1988v043n04ABEH001892.
[14]	Y. Pesin, "Dimension Theory in Dynamical Systems, Contemporary Views and Applications," University of Chicago Press, Chicago, 1997.
[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.
[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.
[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.
[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.
[19]	D. Thompson, Irregular sets, the $\beta-$transformation and the almost specification property, preprint, arXiv:0905.0739v1.
[20]	P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
[21]	P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682.
[22]	G. Zhang, Variational principles of pressure, Discrete Continuous Dynam. Systems A, 24 (2009), 1409-1435.
[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.
[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.

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