In the present paper, we study the thermodynamical properties of finitely generated continuous subgroup actions. We propose a notion of topological entropy and pressure functions that do not depend on the growth rate of the semigroup and introduce strong and orbital specification properties, under which the semigroup actions have positive topological entropy and all points are entropy points. Moreover, we study the convergence and Lipschitz regularity of the pressure function and obtain relations between topological entropy and exponential growth rate of periodic points in the context of semigroups of expanding maps, obtaining a partial extension of the results obtained by Ruelle for ℤd-actions [D. Ruelle, Trans. Am. Math. Soc., 187, 237–251 (1973)]. The specification properties for semigroup actions and the corresponding one for its generators and the action of push-forward maps are also discussed.

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