Pressures for multi-potentials in semigroup dynamics

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 the respective capacities on non-compact sets $Y$, for multi-potentials $\Phi$. This is done by using Carath\'eodory-Pesin structures. Several properties of these types of pressure, and relations between them are explored. The inverse limit of the semigroup and its relations to the above pressures are studied. These notions can be used to classify semigroup actions. We introduce a notion of measure-theoretic amalgamated entropy, and prove a Partial Variational Principle for the amalgamated pressure. 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 the unstable multi-potential to estimate the dimensions of slices in $G$-invariant saddle-type sets.