On a class of automorphisms in H2 which resemble the property of preserving volume

Abstract

AbstractWe give a possible extension of definition of shears and overshears in the case of two non commutative (quaternionic) variables in relation with the associated vector fields and flows. We define the divergence operator and determine the vector fields with divergence. Given the non‐existence of quaternionic volume form on H2, we define automorphisms with volume to be time‐one maps of vector fields with divergence and volume preserving automorphisms to be time‐one maps of vector fields with divergence 0. To these two classes the Andersen–Lempert theory applies. Finally, we exhibit an example of a quaternionic automorphism, which is not in the closure of the set of finite compositions of volume preserving quaternionic shears even though its restriction to the complex subspace is in the closure of the set of finite compositions of complex shears.