Inverse pressure for finitely generated semigroups

We introduce and study an inverse topological pressure P- and its capacities for multi-potentials with respect to the dynamics of finitely generated semigroups G of non-invertible maps (see Mihailescu, 2021) on arbitrary sets Y in compact metric spaces X. These notions are very different from the (forward) pressure notions studied previously and reflect other aspects of the semigroup dynamics. We prove several properties of this inverse pressure. Examples of inverse entropy/pressure for semigroups are also discussed. Then, we consider a G-invariant compact set lambda for a semigroup G of smooth endomorphisms on a manifold M, having a G-stable cone field and a G-unstable cone field over lambda. The inverse pressure of the stable multi-potential is applied to obtain dimension estimates for slices through lambda which are transversal to the unstable directions. (c) 2022 Elsevier Ltd. All rights reserved.