Abstract: The applicability of Bowen’s equation to an arbitrary subset Z of a compact metric space has been well-studied by Climenhaga [V. Climenhaga, Bowens equation in the non-uniform setting, Ergodic Theory Dynam. Sys tems (2011), 31, 1163-1182]. Our work aims to generalize the main results obtained by Climenhaga to free semigroup actions. To this end, we adopt the Carath´eodory-Pesin structure (C-P structure) and introduce the notions of the topological pressure and lower and upper capacity topological pressures of a free semigroup action for an arbitrary subset. Some properties of these notions follow and we obtain the following three main results. First, by Bowen’s equation, we characterize the Hausdorff dimension of an arbitrary subset, where the points of the subset have the positive lower Lyapunov exponents and satisfy a tempered contraction condition. Second, we give an estimation of the topological pressure of a free semigroup action on an arbitrary subset. Finally, we analyze the relationship between the upper capacity topological pressure of a skew-product transformation and a free semigroup action in an arbitrary subset.
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