EQUIVARIANT JACOBIAN CONJECTURE IN DIMENSION TWO

Let G be a small finite subgroup of GL(2,C) and let φ̃ : A → A be a G-equivariant étale endomorphism of the affine plane. We show that φ̃ is an automorphism if the order of G is even. The proof depends on an analysis of a quasi-étale endomorphism φ induced by φ̃ on the singular quotient surface A/G whose smooth part X◦ has the standard A ∗ -fibration p◦ : X◦ → P. If φ preserves the standard A ∗ -fibration p◦ then both φ and φ̃ are automorphisms. We look for the condition with which φ preserves the standard A ∗ -fibration and prove as a consequence that φ̃ is an automorphism if G has even order.