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We study several notions of topological pressure and capacities formulti-potentials $\Phi \in \mathcal C(X;\mathbb R)^m$, with respect to finitelygenerated continuous semigroups $G$ on a compact metric space $X$. We introducethe amalgamated pressure, the condensed pressure, the trajectory pressure, theexhaustive pressure, and the respective capacities on non-compact sets $Y$, formulti-potentials $\Phi$. This is done by using Carath\'eodory-Pesin structures.Several properties of these types of pressure, and relations between them areexplored. The inverse limit of the semigroup and its relations to the abovepressures are studied. These notions can be used to classify semigroup actions.We introduce a notion of measure-theoretic amalgamated entropy, and prove aPartial Variational Principle for the amalgamated pressure. Local amalgamatedentropies and local exhaustive entropies are introduced for probabilitymeasures on X, and we show they provide estimates for the amalgamated and theexhaustive entropies of non-compact sets. Also we find a bound for the localexhaustive entropy for marginal measures on $X$. We apply the amalgamatedpressure of the unstable multi-potential to estimate the dimensions of slicesin $G$-invariant saddle-type sets.

Pressures for multi-potentials in semigroup dynamics