Variational principle for topological pressures on subsets. (English) Zbl 1320.37016
Summary: This paper studies the relations between Pesin-Pitskel topological pressure on an arbitrary subset and measure theoretic pressure of Borel probability measures, which extends D.-J. Feng and W. Huang’ recent result on entropies [J. Funct. Anal. 263, No. 8, 2228–2254 (2012; Zbl 1267.37015)] for pressures. More precisely, this paper defines the measure theoretic pressure \(P_\mu(T, f)\) for any Borel probability measure, and shows that \(P_B(T, f, K)=\sup\{P_\mu(T, f):\mu\in\mathcal M(X),\mu(K)=1\}\), where \(\mathcal M(X)\) is the space of all Borel probability measures, \(K\subseteq X\) is a non-empty compact subset and \(P_B(T, f, K)\) is the Pesin-Pitskel topological pressure on \(K\). Furthermore, if \(Z\subseteq X\) is an analytic subset, then \(P_B(T, f, Z)=\sup\{P_B(T, f, K):K\subseteq Z\text{ is compact}\}\). This paper also shows that Pesin-Pitskel topological pressure can be determined by the measure theoretic pressure.
MSC:
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37B40 | Topological entropy |
Keywords:
measure-theoretic pressure; variational principle; Borel probability measure; topological pressureCitations:
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