Persistence of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings (Q489225)

The article under review investigates a class of nearly integrable symplectic mappings and considers possible persistence of lower-dimensional hyperbolic invariant tori. The setting is the following: There is a symplectic mapping \[ \Phi: \mathbb{T}^n\times W\times O\times W\to \mathbb{T}^n\times \mathbb{R}^m\times \mathbb{R}^n\times \mathbb{R}^m, \] where \(O\) is a closed, bounded domain in \(\mathbb{R}^n\), \(W\) is a small ball centered at the origin in \(\mathbb{R}^n\) and \(\mathbb{T}^n\) is the usual \(n\)-torus. The symplectic mapping arises from a Hamiltonian \(H(x,u,\widehat y,\widehat v)\) on \(\mathbb{T}^n\times W\times O\times W\) with \[ \begin{aligned} \widehat x=\partial_{\widehat y}H(x,u,\widehat y,\widehat v),\quad & y=\partial_x H(x,u,\widehat y,\widehat v),\\ \widehat u=\partial_{\widehat v}H(x,u,\widehat y,\widehat v),\quad & v= 2_uH(x,u,\widehat y,\widehat v),\end{aligned} \] where \(H(x,u,\widehat y,\widehat v)= N(x,u,\widehat y,\widehat v)+ P(x,u,\widehat y,\widehat v)\), \(N(x,u,\widehat y,\widehat v)= \langle x,\widehat y\rangle+ h(\widehat y)+\langle u,A\widehat v\rangle\), \(h:\mathbb{R}^n\to\mathbb{R}\), \(A\) is a non-singular symmetric matrix, \(P\) is a perturbation and \(\langle\cdot,\cdot\rangle\) is the Euclidean inner product. Main result: If \(h\) and \(P\) are real anaytic on \(\mathbb{T}^n\times W\times O\times W\), \(\omega(y)= h'(y)\), \(\widetilde\omega=(1,\omega)\) and \(\widetilde\omega\) satisfies a Rüssmann non-degeneracy condition, so that \(\widetilde\omega\) does not fall on a hyperplane through the origin, then for sufficiently small \(\varepsilon\) and \(|P|<\varepsilon\), there is a non-empty Cantor-like subset \(O_*\subset O\) such that the symplectic mapping defined above has a family of invariant tori \(\{T_\chi\}\) for \(\xi\in O_*\). The reader should be warned that the paper has quite a number of typographical errors and other infelicities.