Smooth circle actions on $S^7$ with unbounded periods and non-linearizable multicentres

Abstract

We will give an example of a smooth free action of $S^1=U(1)$ on $S^7$ whose orbits have unbounded lenghts (equivalently: unbounded periods). As an application of this example we construct a $C^{\infty}$ vector field $X$, defined in a neighbourhood $U$ of $0 \in \mathbb{R}^8$, such that: $U-\{0\}$ is foliated by closed integral curves of $X$, the differential $DX(0)$ at $0$ defines a $1$-parameter group of nondegenerate rotations and $X$ is {\it not} orbitally equivalent to its linearization. This proves in the $C^{\infty}$ category that the classical Poincar\'e Centre Theorem, true for planar nondegenerate centres, is not generalizable to multicentres.