Smooth foliations by circles of 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$X$isnotorbitally 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 islocally unboundednear 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$S^{1}{\hookrightarrow}S^{7}\rightarrow \mathbb{CP}^{3}$.