In this paper, we study the pressure of C1-smooth partially hyperbolic diffeomorphisms with sup-additive potentials. We give several definitions of the so called unstable (measure theoretic) pressure in terms of Bowen's picture and the capacity picture. We show that all such unstable metric pressures of a given ergodic measure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponent of the potentials with respect to the ergodic measure.
Wenda ZHANG
,
Zhiqiang LI
. SUP-ADDITIVE METRIC PRESSURE OF DIFFEOMORPHISMS[J]. Acta mathematica scientia, Series B, 2022
, 42(2)
: 575
-587
.
DOI: 10.1007/s10473-022-0211-z
[1] Bowen R. Equilibrium states and the ergodic theory of Anosov diffeomorphisms//Lect Notes in Math 470. Berlin Heidelberg:Springer-Verlag, 1975
[2] Barreira L. A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergodic Theory Dynam Systems, 1996, 16:871-927
[3] Cao Y, Feng D, Huang W. The thermodynamic formalism for sub-multiplicative potentials. Discrete Contin Dyn Syst, 2008, 20:639-657
[4] Cao Y, Hu H, Zhao Y. Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure. Ergodic Theory Dynam. Systems, 2013, 33:831-850
[5] Falconer K. A subadditive thermodynamic formalism for mixing repellers. J Phys A, 1988, 21:737-742
[6] Huang P, Chen E, Wang C. Katok's entropy formula of unstable metric entropy for partially hyperbolic diffeomorphisms. Preprint. arXiv:1811. 05278v2
[7] He L, Lv J, Zhou L. Definition of measure-theoretic pressure using spanning sets. Acta Math Sinica Engl Ser, 2004, 20:709-718
[8] Hu H, Hua Y, Wu W. Unstable entropies and variational principle for partially hyperbolic diffeomorphisms. Adv Math, 2017, 321:31-68
[9] Hu H, Wu W, Zhu Y. Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphisms. Preprint. arXiv:1710. 02816v1, doi:10. 1017/etds. 2020. 105
[10] Katok A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst Hautes Études SciPubl Math, 1980, 51:137-173
[11] Pesin Ya, Pitskel B. Topological pressure and the variational principle for noncompact sets. Functional Anal Appl, 1984, 18:307-318
[12] Rohlin A V. On the fundamental ideas of measure theory. J Amer Math Soc Translation, 1952, 71:55 pp
[13] Ruelle D. Statistical mechanics on a compact set with Zv action satisfying expansiveness and specification. Trans Amer Math Soc, 1973, 187:237-251
[14] Tian X, Wu W. Unstable entropies and dimension theory of partially hyperbolic systems. Preprint. arXiv:1811. 03797
[15] Walters P. A variational principle for the pressure of continuous transformations. Amer J Math, 1975, 97:937-971
[16] Walters P. An introduction to ergodic theory. Graduate Texts in Mathematics. 79. New York-Berlin:Springer-Verlag, 1982
[17] Zhang W, Li Z, Zhou Y. Unstable topological pressure of partially hyperbolic diffeomorphisms with subadditive potentials. Preprint. arXiv:2008. 00454v2
[18] Zhang W, Li Z, Zhou Y. Unstable metric pressure of partially hyperbolic diffeomorphisms with sub-additive potentials. Nonlinearity, 2020, 30:6915-6934
[19] Zhao Y, Cao Y. Measure theoretic pressure for subadditive potentials. Nonlinear Analysis, 2009, 70:2237- 2247
[20] Zhao Y. A note on the measure theoretic pressure in subadditive case. Chinese Annals of Math Series A, 2008, 3:325-332
