On the dynamics of some diffeomorphisms of \(\mathbb{C}^2\) near parabolic fixed points (Q2705780)

This paper deals with diffeomorphisms \(F: \mathbb{C}^2\to\mathbb{C}^2\) of the following special form (1): \(F(z,w)= (w,-z+ 2G(w))\), where \(G\in C^1(\mathbb{C})\) is holomorphic near \(0\), \(G(0)= 0\), \(G'(0)= 1\). Then \(F(0,0)= (0,0)\), the eigenvalues of \(F'(0, 0)\) are both equal to \(1\) and \(F'(0, 0)\) is nondiagonalizable. For such mapping \(F\), the authors have proved that there exists a function \(f\) univalent in a convex domain \(\Omega\subset\mathbb{C}\), such that \(\Omega\subset f(\Omega)\) and the functions \(f\) and \(g= f^{-1}\) have the following property: the graphs \(\{(z, g(z)): z\in\Omega\}\) and \(\{(z, f(z)): z\in\Omega\}\) of \(g\) and \(f\) are invariant under of \(F\) and \(F^{-1}\) respectively, and \(F^n(z, g(z))\to 0\), \(F^{-n}(z, f(z))\to 0\), as \(n\to\infty\), locally uniformly for \(z\in\Omega\). Further, the special choice of the function \(G(z)= z+ az^{j-1}+ \sum_{k\geq j+2} b_kz^k\), \(a>0\), \(j\geq 1\), \(b_k\in \mathbb{R}\), is investigated.