Klein's conjecture for contact automorphisms of the three-dimensional affine space

Abstract

where ρ(x, y,p) is a nonzero rational function. Our reference to a “plane” may seem mistaken, but classically it means an action of the transformation on plane contact elements ((x, y),p) consisting of a point (x, y) and a slope p at the point. We will say that ρ(x, y,p) is the multiplier of T. The contact transformation T is said to be a contact affine transformation (or contact polynomial automorphism) if T and its inverse T −1 are polynomial. For a contact affine transformation of A3, the multiplier is a nonzero constant (see Lemma 2.1 for contact transformations of odd-dimensional spaces).