Topological Pressure For Sub-additive Potentials Of Accountable Discrete Amenable Actions

We define a local topological pressure for sub-additive potentials of countable discrete amenable group actions, and prove the corresponding local variational principle. More precisely, for a given dynamical system (X, G) where G is a countable discrete amenable group, an finite open covering U of X, and a sub-additive potential on (X,G) F={fE∈C(X):E∈F(G)}, under certain conditions, we show that the corresponding local pressure P(G,F;U) satisfies P(G,F;U)= sup {hμ(G,U)+F*(μ)},μ∈M(X,G) moreover, the supremum can be attained by a G-invariant ergodic measure. For this kind of generalization, we present a nontrivial example. As applications,by considering the local pressure relative to a special additive potential which is induced from a fixed continuous function, we obtain some results which show that local pressures determine invariant measures and local measure-theoretic entropies. Besides, the properties of local equilibrium states are studied.