Reversible Polynomial Automorphisms of the Plane: the Involutory Case

Planar polynomial automorphisms are polynomial maps of the plane whose inverse is also a polynomial map. A map is reversible if it is conjugate to its inverse. Here we obtain a normal form for automorphisms that are reversible by an involution that is also in the group of polynomial automorphisms. This form is a composition of a sequence of generalized Henon maps together with two simple involutions. We show that the coefficients in the normal form are unique up to finitely many choices.