Gevrey regularity of invariant curves of analytic reversible mappings
More than a decade ago, it was discovered by \textit{G. Popov} [Ann. Henri Poincaré 1, No. 2, 223--248 (2000; Zbl 0970.37050)] that KAM tori of real-analytic Kolmogorov non-degenerate nearly integrable Hamiltonian systems form a Gevrey-regular family (in the sense of Whitney). The result of the present paper lies in this area of research. The authors consider a real-analytic family of real-analytic maps of the annulus \(A_p(x,z;p)=(x_1,z_1;p)\), \(x_1=x+\omega(p) + f(x,z;p)\), \(z_1=z + g(x,z;p)\), reversible with respect to the involution \((x,y)\to (-x,y)\). Here \(x\in \mathbb R/2\pi \mathbb Z\) and \(z\in \mathbb R\) are the phase variables, and \(p\in \Pi\subset \mathbb R\) is a parameter, where \(\Pi\) is a bounded interval containing 0. It is assumed that \(\omega^{(j)}(0)=0\), \(0\leq j\leq m\), \(\omega^{(m)}(0)\neq 0\). Omitting all quantitative details, the main result can be presented as follows: Theorem: Fix arbitrary numbers \(\tau\geq m-1\), \(\tau+2< \mu <\tau+3\) and \(0<\alpha <1\). Then, for \(f\) and \(g\) sufficiently small, there exists a Cantor set \(\Pi_*\subset \Pi\) and a family of transformations \(x=\xi+p_*(\xi; p)\), \(y=\eta +q_*(\xi,\eta; p)\), defined for \(p\in \Pi_*\) and close to identity, with the following properties. The functions \(p_*\) and \(q_*\) are real-analytic in \((\xi,\eta)\) and Gevrey in \(p\) (in the sense of Whitney) with the Gevrey index \(\mu\). In the new coordinates, the family \(A_p\) takes the form \(\xi_1=\xi+\omega_*(p)+f_*(\xi,\eta;p)\), \(\eta_1=\eta + g_*(\xi,\eta;p)\), where \(f_*=O(\eta)\), \(g_*=O(\eta^2)\). The number \(\omega_*(p)\) is Diophantine: \(|k\omega_*(p)-2\pi l| \geq \pi\alpha |k|^{-\tau}\) for all non-zero integers \(k\), \(l\). Moreover, the Lebesgue measure of \((\Pi \setminus \Pi_*)\) does not exceed \(c\alpha^{1/m}\). Thus, for any \(p\in \Pi^*\), the mapping \(A_p\) admits a real-analytic invariant curve with rotation number \(\omega_*(p)\), and these curves depend on \(p\) in a Gevrey-regular way. The proof follows the standard KAM procedure. The result can be applied to individual reversible mappings of the form \((x,y)\mapsto (x_1,y_1)\), \(x_1=x+h(y)+F(x,y)\), \(y_1=y+G(x,y)\). Reviewer's remark: The introduction, and even the title of this paper, are a little misleading. A more suitable title could be ``Gevrey regularity of families of invariant curves of analytic reversible mappings''.
scientific article; zbMATH DE number 5847734