A necessary and sufficient condition for a twist map being integrable
The paper deals with an area preserving monotone twist diffeomorphism \(T: S^1 \times \mathbb{R} \to S^1 \times \mathbb{R}\). Let us choose a point \((x_0, y_0) \in \mathbb{R}^2\) and consider the trajectory \((x_n,y_n) = T^n(x_0, y_0)\), \(n \in \mathbb{Z}\). The sequence \(\{x_n\}\) is called a configuration. The authors firstly introduce the notion of Jacobi field: A sequence \(\{\xi_n\}\) is called Jacobi field along the configuration \(\{x_n\}\) if the following equations are satisfied for all \(n\in \mathbb{Z}\): \[ h_{12} (x_{n-1}, x_n) \xi_{n-1} + \bigl[h_{22} (x_{n-1}, x_n) + h_{11} (x_n,x_{n+1}) \bigr] \xi_n + h_{12} (x_n, x_{n+1}) \xi_{n+1} =0, \] where \(h: \mathbb{R}^2 \to \mathbb{R}\) is the generating function of \(T\) and \(h_{11}, h_{12}, h_{22}\) are the corresponding second order partial derivatives of \(h\) with respect to the first and second variable. Two points \(x_M\), \(x_N\) of a configuration are called conjugate if there is a nonzero Jacobi field vanishing at \(x_M\), \(x_N\). An area preserving monotone twist diffeomorphism is said to be one without conjugate points if there are no conjugate points along any configurations. Several properties of Jacobi fields are investigated. One of the important results is: Let \(\{x_n\}\) and \(\{x^*_n\}\) be two configurations of the map \(T\). There is a sequence of \(C^1\) functions \(\{f_n (t)\}\) defined on \([0,1]\) such that \(f_n(0) = x_n\), \(f_n(1) = x^*_n\) and for each \(t_0 \in [0,1]\), \(\{f_n (t_0)\}\) is a configuration of the map \(T\), and \(\{f_n' (t_0)\}\) is a Jacobi field along this configuration. Finally, the main result of the paper states: For an area preserving monotone twist diffeomorphism \(T\) the phase space \(\Omega = S^1 \times \mathbb{R}\) is foliated by continuous closed invariant curves not null homotopic on \(\Omega\) if and only if \(T\) has no conjugate points. This interesting paper is well written and organized.
scientific article; zbMATH DE number 940881