Continuity of Julia sets (Q1297628)

The author discusses the problem of continuity of Julia sets in the family of rational maps. The following theorem is proved. Let \(\mathcal R_k\) be the family of all rational maps of degree \(k > 1.\) Then the Julia set \(J(R)\) is continuous at \(R_0\) iff \(R_0\) has no parabolic, Siegel periodic point or Herman ring. For a general analytic family of rational maps the similar result to the conclusion of this theorem may not hold. The following assertion takes place. Let \(R_w(z):W\times\overline C\to\overline C\) be an analytic family of rational maps of degree \(k>1.\) Then \(J(w)\) is continuous at \(w_0\in W\) if each parabolic point of \(R_{w_0}\) is strongly persistently nonhyperbolic and each Siegel periodic point of \(R_{w_0}\) is persistently nonhyperbolic and each Herman ring in \(F(w_0)\) is structurally stable. Here \(F(R)=\overline C\setminus J(R)\) is the Fatou set.