Local connectivity of the Julia set for geometrically finite rational maps (Q1919035)

The authors call a rational map \(f\) of \(\mathbb{C}\) of degree \(d>1\) geometrically finite if every critical point is either eventually periodic or attracted to an attracting or parabolic periodic cycle. The paper shows that if the Julia set \(J\) of such a map is connected, then \(J\) is also locally-connected. This was alaready known in the special cases when \(f\) is a polynomial or a hyperbolic rational map. In the case of a general geometrically finite rational map with non-connected \(J\) there remains a conjecture that every component of connectivity of \(J\) is locally connected. Combining the main theorem of the paper with a result of McMullen shows at least that every eventually periodic component of \(J\) is locally connected.