  We study several notions of topological pressure and capacities for
multi-potentials $\Phi \in \mathcal C(X;\mathbb R)^m$, with respect to finitely
generated continuous semigroups $G$ on a compact metric space $X$. We introduce
the amalgamated pressure, the condensed pressure, the trajectory pressure, the
exhaustive pressure, and respective capacities on non-compact sets, for
multi-potentials $\Phi$. This is done by using Caratheodory-Pesin structures.
Several properties of these types of pressure, and relations between them are
explored. We introduce a notion of measure-theoretic amalgamated entropy, and
prove a Partial Variational Principle for amalgamated pressure. The inverse
limit of the semigroup and its relations to these pressures are studied. Local
amalgamated entropies and local exhaustive entropies are introduced for
probability measures on X, and we show they provide estimates for the
amalgamated and the exhaustive entropies of non-compact sets. Also we find a
bound for the local exhaustive entropy for marginal measures on $X$. We apply
the amalgamated pressure of unstable multi-potentials, in order to estimate the
Hausdorff dimension of unstable intersections with $G$-invariant hyperbolic
sets.
