We introduce and explore new notions of pressure of multipotentials \(\Phi = (\varphi _j)_{j \in \Omega }\) along arbitrary sets of trajectories \(\mathcal Y \subset \Sigma _\Omega ^+ \times X\) for semigroups generated by a set \(G_1 = (f_j)_{j \in \Omega }\) of continuous maps on a compact metric space X. The most useful among them is the amalgamated pressure \(P^A(\Phi , \mathcal Y, G_1)\) . Amalgamated pressure is applied to formulas or estimates for Hausdorff dimension. First if \(G_1\) is a finite set of conformal maps on a metric space X, we obtain a formula for HD(Y) for sets Y on which the \(G_1\) -Lyapunov exponents are positive. In particular, this gives a formula for \(HD(J_f \cap J_g)\) if \(J_f, J_g\) are conformal repellers in \(\mathbb R^k\) with non-empty intersection. This applies also to semigroups generated by parabolic maps. We obtain the dimension of the projection set \(G_\mu \) of the set \(\mathcal G_\mu \) of generic trajectories for a \(G_1\) -expanding measure \(\mu \) on \(\Sigma _\Omega ^+ \times X\) . Next we study the case when \(G_1\) is a finite set of \(\mathcal C^2\) maps on a manifold M and there exist \(G_1\) -partial stable and unstable cone fields on a compact set \(\Lambda \) . For a non-uniformly hyperbolic subset \(Y \subset \Lambda \) we apply the amalgamated pressure of the unstable multipotential \(\Phi ^u\) , to estimate the Hausdorff dimension of the slices through Y with arbitrary submanifolds transversal to stable cones. We introduce then a measure-theoretic amalgamated entropy \(h^A(\mu , G_1)\) for the marginal measure of any F-invariant ergodic measure \(\mu \) on \(\Sigma _\Omega ^+ \times X\) , and prove a partial Variational Principle for the amalgamated pressure. Semigroups of toral endomorphisms are studied also, and computations are provided. We obtain also a new notion of pressure for random dynamical systems and random potentials.
