Hyperbolic automorphisms and holomorphic motions in \({\mathbb C}^2\)
The authors establish analogs of results of \textit{R. Mañé, P. Sad} and \textit{D. Sullivan} [Ann. Sci. Éc. Norm. Supér. (4) 16, 193-217 (1983; Zbl 0524.58025)]. One of their main results is: Theorem 1.1. Let \(f_\lambda\) be a 1-parameter family of hyperbolic Hénon maps depending holomorphically on \(\lambda\in \Delta^n\). Then there exists \(r>0\) and a map \[ \psi: \Delta^n_r \times(I^+_0\cup I^-_0)\to I^+_\lambda\cup I^-_\lambda \] such that, defining \(\psi_\lambda(p) =\psi(\lambda,p)\), we have: 1) \(\psi_0(p)= p\): 2) \(\psi_\lambda\) is a homeomorphism for each fixed \(p\in I^+_0\cup I^-_0\); 3) \(\psi_\lambda(p)\) is holomorphic in \(\lambda\) for each fixed \(p\in I^+_0\cup I^-_0\); 4) \(\psi_\lambda\) maps each leaf of \(I^-_0(I^+_0)\) to a leaf of \(I^-_\lambda (I^+_\lambda)\); and 5) \(\psi_\lambda f_0=f_\lambda \psi_\lambda\) on \(I^+_0\cup I^-_0\).
scientific article; zbMATH DE number 1742560