Hénon mappings with biholomorphic escaping sets

For any complex Hénon map H P , a : x y ↦ P ( x ) - a y x , the universal cover of the forward escaping set U + is biholomorphic to D × C , where D is the unit disk. The vertical foliation by copies of C descends to the escaping set itself and makes it a rather rigid object. In this note, we give evidence of this rigidity by showing that the analytic structure of the escaping set essentially characterizes the Hénon map, up to some ambiguity which increases with the degree of the polynomial P .