KAM theorem for reversible mapping of low smoothness with application
KAM theorem for reversible mapping of low smoothness with application
Assume the mapping$ A:\left\{ \begin{array}{ll} x_{1} = x+\omega+y+f(x,y),\\ y_{1} = y+g(x,y), \end{array} \right. (x, y)\in \mathbb{T}^{d}\times B(r_{0}) $is reversible with respect to $ G: (x, y)\mapsto (-x, y), $ and $ | f | _{C^{\ell}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon_{0}, | g |_{C^{\ell+d}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon_{0}, $ where $ B(r_{0}): = \{|y|\le r_0:\; y\in\mathbb …