# American Institute of Mathematical Sciences

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

## Pressures for asymptotically sub-additive potentials under a mistake function

 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.
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.

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##### 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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