Local \(C^{\infty}\) conjugacy on the Julia set for some holomorphic perturbations of \(z\to z^ 2\)
Denote by \(f_ 0\) the map of the complex plane given by \(z\to z^ 2\) and set \(f(z)=z^ 2+r(z)\) where r(z) is analytic in the annulus \(C(\alpha)=\{z: e^{-\alpha}\leq | z| \leq e^{\alpha}\}\). The Julia set J of \(f_ 0\) is the unit circumference which contains all the unstable periodic orbits. Denote the set of periodic orbits of f by \(\Gamma\) (f) and the multiplier of a member \(\gamma\) of this set by \(Sp(f)(\gamma)=(f^ n)'(z),\quad z\in \gamma.\) Write \(\| r\|_{\alpha}\) for sup\(| r(z)|\) in C(\(\alpha)\). The main result is the following theorem. Given \(\alpha >0\), there is a number \(\epsilon (\alpha)>0\) such that if \(\| \Gamma \|_{\alpha}<\epsilon (\alpha)\), then there is a homeomorphism h defined on J such that \(\Gamma_{f| C_{\alpha}}=h(\Gamma_{f_ 0| C_{\alpha}}).\) If moreover \[ Sp(f)\circ h|_{\Gamma_{(f_ 0| C_{\alpha})}}=Sp(f_ 0)|_{\Gamma (f_ 0| C_{\alpha})} \] then h is a \(C^{\infty}\)-diffeomorphism such that \(h\circ f_ 0=f\circ h\). Further, if \(\| r\|_{\alpha}\to 0\) then \(h\to Id\) in \(C^ p\) for any p.
scientific article; zbMATH DE number 3889876