We prove that a nilpotent subgroup of orientation preserving $C^{ 1}$ diffeomorphisms of ${ \mathbb S}^{ 2}$ has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation preserving $C^{ 1}$ diffeomorphisms of ${ \mathbb R}^{ 2}$ preserving a compact set has a global fixed point. These results generalize theorems of Franks, Handel and Parwani for the abelian case. We show that a nilpotent subgroup of orientation preserving $C^{ 1}$ diffeomorphisms of ${ \mathbb S}^{ 2}$ that has a finite orbit of odd cardinality also has a global fixed point. Moreover we study the properties of the two-points orbits of nilpotent fixed-point-free subgroups of orientation preserving $C^{ 1}$ diffeomorphisms of ${ \mathbb S}^{ 2}$.
Categories: math.DS