Algebraic Vector Fields on $S^2$ are $2$-generated

Abstract

We exhibit two vector fields on $S^2$, which generate the entire Lie algebra of algebraic vector fields. Compositions of the flows associated with these two vector fields enable us to approximate every diffeomorphism of $S^2$ isotopic to the identity in the ${\mathrm{C}}^r$-sense for any $r\ge 0$.