Abstract
We define the relative local topological pressure for any given factor map and open cover, and prove the relative local variational principle of this pressure. More precisely, for a given factor map π: (X, T) → (Y,S) between two topological dynamical systems, an open cover U of X, a continuous, real-valued function f on X and an S-invariant measure ν on Y, we show that the corresponding relative local pressure P(T, f, U, y) satisfies
, where M(X, T) denotes the family of all T-invariant measures on X. Moreover, the supremum can be attained by a T-invariant measure.
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Blanchard F. A disjointness theorem involving topological entropy. Bull Soc Math France, 1993, 121: 465–478
Blanchard F, Glasner E, Host B. A variation on the variational principle and applications to entropy pairs. Ergodic Theory Dynam Systems, 1997, 17: 29–43
Blanchard F, Host B, Maass A, et al. Entropy pairs for a measure. Ergodic Theory Dynam Systems, 1995, 15: 621–632
Bourbaki N. Integration, Chapter VI. Paris: Hermann, 1959
Castaing C, Valadier M. Convex Analysis and Measurable Multifunctions. In: Lecture Notes in Math, vol. 580. New York: Springer-Verlag, 1977
Glasner E, Weiss B. On the interplay between measurable and topological dynamics. In: Handbook of Dynamical Systems, vol. 1B. Amsterdam: Elsevier, 2006, 597–648
Huang W, Maass A, Romagnoli P, et al. Entropy pairs and a local abramov formula for a measure theoretical entropy of open covers. Ergodic Theory Dynam Systems, 2004, 24: 1127–1153
Huang W, Ye X. A local variational relation and applications. Israel J Math, 2006, 151: 237–280
Huang W, Ye X, Zhang G. A local variational principle for conditional entropy. Ergodic Theory Dynam Systems, 2006, 26: 219–245
Huang W, Yi Y. A local variational principle for pressure and its applications to equilibrium states. Israel J Math, 2007, 161: 29–74
Ledrappier F, Walters P. A relativised variational principle for continuous transformations. J London Math Soc (2), 1977, 16: 568–576
Parry W. Topics in Ergodic Theory. In: Cambridge Tracts in Mathematics, no. 75. New York: Cambridge University Press, 1981
Romagnoli P. A local variational principle for the topological entropy. Ergodic Theory Dynam Systems, 2003, 23: 1601–1610
Ruelle D. Statistical mechanics on a compact set with Zυ action satisfying expansiveness and specification. Trans Amer Math Soc, 1973, 187: 237–251
Walters P. A variational principle for the pressure of continuous transformations. Amer J Math, 1975, 97: 937–971
Walters P. An Introduction to Ergodic Theory. New York: Springer-Verlag, 1982
Ye X, Huang W, Shao S. Introduction to Topological Dynamical Systems (in Chinese). Beijing: Science Press, 2008
Zhang G. Relativization and localization of dynamical properties. PhD Thesis. Hefei: University of Science and Technology of China, 2007
Zhang G. Variational principles of pressure. Discrete Contin Dyn Syst, 2009, 24: 1409–1435
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Ma, X., Chen, E. & Zhang, A. A relative local variational principle for topological pressure. Sci. China Math. 53, 1491–1506 (2010). https://doi.org/10.1007/s11425-010-3038-3
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DOI: https://doi.org/10.1007/s11425-010-3038-3