Topological Pressure Of Free Semigroup Actions On Noncomapct Subsets, Bowen’s Equation And Variational Principle

Topological entropy and topological pressure are important concepts of dynamical systems,characterizing their complexity.In recent years,there has been increasing interest in dynamical systems with group or semigroup actions.Numerous pioneering results have emerged and significantly enriched related research fields.Dynamical systems with group or semigroup actions are more complex than their classical counterparts.For dynamical systems with free semigroup actions,we introduce two different notions of topological pressures for noncompact sets using the Carathéodory–Pesin structure,and study their properties and their corresponding Bowen’s equations.We give several applications of the Bowen’s equation and prove the variational principle for the second topological pressure.More precisely,1)For the first topological pressure,several properties are given and the following main results are proved:· We use Bowen’s equation of two topological pressures to study the Hausdorff dimension of a set satisfying certain conditions.· We estimate these topological pressures on a subset using the local measure pressure.· We give the relation between the upper capacity topological pressures of free semigroup actions,and the upper capacity topological pressures of skew-product transformations.2)For the second topological pressure,several properties and the following applications are also given:· We use Bowen’s equation of two topological pressures to study the Hausdorff dimension of a set satisfying certain conditions.· We estimate these topological pressures on a subset using the local measure pressure.· We reveal the relation between measure pressures and topological pressures,namely,the variational principle.