Existence of invariant tori for certain weakly reversible mappings (Q1565987)

The author considers the mapping \({\mathcal M}\) close to an integrable one \(x_1=x+h(y)+f(x,y)\), \(y_1=y+g(x,y)\) and close to an involution \(x_1=-x+p(x,y)\), \(y_1=y+q(x,y)\) where \((x,y)\in T^m\times O(\Omega)\), \(\Omega\in \mathbb{R}^n\) is the closure of a connected bounded open domain and \(O(\Omega)\) is some neighborhood of \(\Omega,f(x,y),g(x,y)\) and \(p(x,y),q(x,y)\) are perturbations. The mapping \({\mathcal M}\) is weakly reversible with respect to the mapping \({\mathcal G}\), i.e. \({\mathcal M}{\mathcal G}{\mathcal M}={\mathcal G}\). Under some weak non-degeneracy condition a uniform KAM iteration is constructed for the proof of the existence of a Cantor family of \(m\)-tori invariant under the reversible mapping \({\mathcal M}\) and the reversing mapping \({\mathcal G}\).