Volume-preserving automorphisms of C⊃n⊃

Abstract

Let Aut₁(C ⁿ ) be the group of all automorphisms F of Cⁿ with Jacobian J(F) equal to 1. A shear is a map of type where f is independent of z_n and Gⁿ is the subgroup of Aut1 (C ⁿ )consisting of all finite compositions of shears. We prove THEOREM A . THLOREM B G ₂ is a proper subgroup of Aut₁(C ²). In particular, the map of C ² is not in G ₂. THEOREM C G _n is dense in Aut₁(C ⁿ) in the topology of uniform convergence on compact subsets. THEOREM D If F:C ⁿ →C ⁿ .Fis entire. J(F)=1 and m∈N, there is S _m∈G _n such that .