Right-handed vector fields and the Lorenz attractor
Studying the celebrated Lorenz system \[ \begin{aligned} \frac{dx}{dt} & =10\left( y-x\right) ,\\ \frac{dx}{dt} & =28x-y-xz,\\ \frac{dx}{dt} & =xy-\frac{8}{3}z \end{aligned} \] from the topological point of view, \textit{J. S. Birman} and \textit{R. F. Williams} [Topology 22, 47--82 (1983; Zbl 0507.58038)] established that the orbits of this system accumulate on some fractal compact invariant set containing a dense countable collection of periodic orbits, each defining a knot referred to in the paper under review as a Lorenz knot. One of the remarkable properties of Lorenz knots is that they are fibered, as well as Lorenz links. The purpose of the paper is to suggest a global point of view at all these fibrations by introducing a concept of right-handed vector fields and analyzing their dynamics. A vector field \(X\) on the 3-sphere is said to be right-handed if the quadratic linking form is positive on the convex set of invariant probability measures. The main result in the paper asserts that, for a right-handed vector field \(X\) in the 3-sphere, any finite collection of periodic orbits is a fibered link. More precisely, any finite collection of periodic orbits is the binding of some Birkhoff section. It is demonstrated that the Lorenz equation cannot be right-handed since it contains a singular point and two trivial unlinked periodic orbits, but it contains arbitrarily large closed invariant subsets which are also invariant subsets of right-handed vector fields in the 3-sphere. For the same reason, the modular flow cannot be right-handed because it has fixed points. Furthermore, modular knots are isotopic to Lorenz knots, and modular links are isotopic to Lorenz links. Consequently, both Lorenz flow and modular flow are almost right-handed in the sense that the linking form is positive on the convex set of invariant measures concentrated on any of the invariant sets \(M_{\varepsilon}\subset\mathbf{S}^{3}\).
scientific article; zbMATH DE number 5644707