In this paper we introduce the notion of scale pressure and measure theoretic for amenable group actions. A variational principle actions is presented. We also describe these quantities by pseudo-orbits. Moreover, prove that if $ G a finitely generated countable discrete group, then coincides with respect to

We establish basic properties of a sheaf graded algebras canonically associated to every relative affine scheme $$f:X\rightarrow S$$ endowed with an action the additive group $${\mathbb {G}}_{a,S}$$ over base or algebraic space S, which we call (relative) Rees algebra -action. In case varieties defined field characteristic zero, further {G}}_{a}$$ -action in terms its locally nilpotent derivati...