On algebraic automorphisms and their rational invariants

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism Φ, we denote by k(X)Φ its field of invariants, i.e., the set of rational functions f on X such that f º Φ = f. Let n(Φ) be the transcendence degree of k(X)Φ over k. In this paper we study the class of automorphisms Φ of X for which n(Φ) = dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form Φ = ϕg, where ϕ is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(Φ) = 1.