Böttcher coordinates (Q2866929)

A germ of a holomorphic function \(f: (\mathbb{C},0)\to (\mathbb{C},0)\) with a superattracting fixed point at the origin can be locally written as NEWLINE\[NEWLINE f(z)=z^k+O(z^{k+1}),\quad k\geq 2, NEWLINE\]NEWLINE and a classical result of Böttcher ensures that there exists a local holomorphic change of coordinate \(\varphi\), tangent to the identity at \(0\), conjugating \(f\) to the map \(w\mapsto aw^k\). The germ \(\varphi\) is called a \textit{local Böttcher coordinate for \(f\)}.NEWLINENEWLINEIn several variables, as observed by \textit{J. H. Hubbard} and \textit{P. Papadopol} [Indiana Univ. Math. J. 43, No. 1, 321--365 (1994; Zbl 0858.32023)], such a result cannot be true in general for a holomorphic germ \(F: (\mathbb{C}^m,0)\to (\mathbb{C}^m,0)\) with a superattracting fixed point at the origin (\(m\geq 2\)). \smallskip In the paper under review, the authors study germs with an \textit{adapted} superattracting fixed point and give necessary and sufficient conditions for the existence of Böttcher coordinates for such germs. Set \(m=m_1+ \cdots + m_p\), with \(m_j \geq 1\). The germ \(F\) has an \textit{adapted superattracting fixed point at \(0\)} if there exist local coordinates \(z=(z_1,\dots,z_p)\) in a neighborhood of \(0\), where \(F=(F_1,\dots,F_p)\) can be written as NEWLINE\[NEWLINE F_j(z)=H_j(z_j)+O(\|z\|^{k_j+1}),\quad k_j\geq2, NEWLINE\]NEWLINE with \(H_j: \mathbb{C}^{m_j} \to \mathbb{C}^{m_j}\) a nondegenerate homogeneous map of order \(k_j\). The map \((H_1, \dots H_p)\) is called the \textit{quasihomogeneous} part of \(F\) at \(0\), and \((k_1,\dots, k_p)\) is its \textit{multidegree}. A \(p\)-tuple of vector fields \((\xi _1, \dots, \xi_p)\) in \(\mathbb{C}^m\) is \textit{admissible} if \(\xi_j\) is tangent to \(\mathbb{C}^{m_j}\), \(\xi_j = \theta_j + o(\|z\|)\), where \(\theta_j\) is the radial vector field in \(\mathbb{C}^{m_j}\), and the vector fields commute pairwise. The main theorem of the paper relates the existence of Böttcher coordinates to the existence of particular admissible \(p\)-tuples of vector fields. More precisely, the authors prove the following.NEWLINENEWLINETheorem 0.3. \textsl{Let \(F: (\mathbb{C}^m,0)\to (\mathbb{C}^m,0)\) be a germ of an analytic map having an adapted superattracting fixed point at \(0\in \mathbb{C}^m\). Let \(H: \mathbb{C}^m\to \mathbb{C}^m\) be the quasihomogeneous part of \(F\) at \(0\) with multidegree \((k_1,\dots, k_p)\). Then the following are equivalent: \smallskip\noindent (1) There is a germ of an analytic map \(\Phi: (\mathbb{C}^m,0)\to (\mathbb{C}^m,0)\) such that NEWLINE\[NEWLINE \Phi(0) = 0,\quad D_0\Phi = id,\quad\text{and}\quad\Phi\circ F = H\circ \Phi. NEWLINE\]NEWLINE \smallskip\noindent (2) There is an admissible \(p\)-tuple of germs of vector fields \((\xi_1, \dots, \xi_p)\) such that NEWLINE\[NEWLINE DF \circ \xi_j =k_j \cdot \xi_j \circ F NEWLINE\]NEWLINE near \(0\) for all \(j\in [1,p]\). \smallskip\noindent (3) There is an admissible \(p\)-tuple of germs of vector fields \((\zeta_1, \dots, \zeta_p)\) such that \(\zeta_j\) is tangent to the germ of the postcritical set of \(F\) for all \(j\in [1,p]\). } \smallskipNEWLINENEWLINEThey also study global Böttcher coordinates in the immediate basin of the superattracting fixed point, proving the following result.NEWLINENEWLINETheorem 0.4. \textsl{Let \(\Omega\) be a complex analytic manifold, \(F: \Omega\to \Omega\) be a proper analytic map with a superattracting fixed point at \(a\in \Omega\), and \(H: T_a\Omega\to T_a\Omega\) be a quasihomogeneous map. Suppose that \smallskip\noindent (1) there is a local isomorphism \(\Phi: (\Omega, a)\to(T_a\Omega, 0)\) with \(\Phi\circ F = H\circ \Phi\); \smallskip\noindent (2) near \(a\), \(\Phi\) maps the postcritical set of \(F:\mathcal{B}_a(F)\to \mathcal{B}_a(F)\) to the postcritical set of \(H:\mathcal{B}_0(H)\to \mathcal{B}_0(H)\). \smallskip Then \(\Phi\) extends to a global isomorphism \(\Phi:\mathcal{B}_a(F)\to \mathcal{B}_0(H)\) conjugating \(F\) to \(H\). } \smallskipNEWLINENEWLINEThe authors present as well various applications and further questions.