Smooth foliations by circles of $S^{7}$ with unbounded periods and nonlinearizable multicentres

Abstract

We give an example of 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 one-parameter group of non-degenerate rotations and $X$ is not orbitally equivalent to its linearization. Such a vector field $X$ has the first integral $I(x)=\Vert x\Vert ^{2}$, and its main feature is that its period function is locally unbounded near the stationary point. This proves in the $C^{\infty }$ category that the classical Poincaré centre theorem, true for planar non-degenerate centres, is not generalizable to multicentres. Such an example is obtained through a careful study and a suitable modification of a celebrated example by Sullivan [A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46 (1976), 5–14], by blowing up the stationary point at the origin and through the construction of a smooth one-parameter family of foliations by circles of $S^{7}$ whose orbits have unbounded lengths (equivalently, unbounded periods) for each value of the parameter and which smoothly converges to the Hopf fibration $S^{1}{\hookrightarrow}S^{7}\rightarrow \mathbb{CP}^{3}$.