Measure-theoretic Pressure Under Z~d-action

Let T be a continuousZd-action on a compact metric space (X, ρ) with d≥1,Ψ: X�'Ra continuous function, and μ a T-invariant measure on X. Modifying thedefinition of measure-theoretic pressure underZ-action, this paper defines the measure-theoretic pressure, the lower and upper capacity measure-theoretic pressure of T withrespect to by using spanning sets and open covers respectively. This paper firstestablishes the Brin-Katok’s entropy formula underZd-action. Then this study showsthat the diferent versions of measure-theoretic pressure defined in this paper are allequivalent. Furthermore, we obtain an inverse variational principle of the measure-theoretic pressure. Precisely, for any T-invariant ergodic measure μ, we construct acertain non-compact set on which the topological pressure and the lower and uppercapacity topological pressure of T are equal to the measure-theoretic pressure of Twith respect to μ.