Hyperbolic automorphisms and holomorphic motions in \({\mathbb C}^2\). (English) Zbl 0993.37024
The authors establish analogs of results of R. Mañé, P. Sad and 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 \(I^+_0\cup I^-_0\).
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 \(I^+_0\cup I^-_0\).
Reviewer: Viorel Vâjâitu (Bucureşti)
MSC:
| 37F15 | Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems |
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
