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Xiao, Qian; Ma, Dongkui

Variational principle of topological pressure of free semigroup actions for subsets. (English) Zbl 1502.37045

Qual. Theory Dyn. Syst. 21, No. 4, Paper No. 99, 18 p. (2022).
Summary: In this paper, we investigate the relation between topological pressure of free semigroup actions for non-compact sets proposed by Q. Xiao and D. Ma [J. Dynam. Differ. Equ. (2021), doi:10.1007/s10884-021-10055-9] and measure-theoretic pressure of Borel probability measure. For any Borel probability measure, this paper defines lower and upper measure-theoretic pressures. Moreover, we give a lower and an upper estimations of the topological pressure of free semigroup actions by local measure-theoretic pressure. In addition, we also show that the topological pressure on a non-empty compact subset \(K\) defined in [loc. cit.] equals to the supremum of the lower measure-theoretic pressure taken over all probability measures supported on \(K\).

Cited in 5 Documents

MSC:

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37B40 Topological entropy
37C45 Dimension theory of smooth dynamical systems
28D20 Entropy and other invariants
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37A05 Dynamical aspects of measure-preserving transformations

Keywords:

variational principle; free semigroup actions; topological pressure; measure-theoretic pressure
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References:

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