Local connectivity and Julia sets. From rational to transcendental maps. Part IV [ Back ]

IMUB-Universitat de Barcelona
Xavier Jarque
Universitat de Barcelona


It is known that local connectivity of Julia sets of polynomials has been a key tool to describe the dynamics. The connection is given by the landing of the external rays associated to the immediate basin of attraction of infinity. Also local connectivity has been extensively studied on the rational scenario. For instance one can prove that if the Julia set of a hyperbolic rational map is connected then it is locally connected. To prove this, one uses Whyborn's Theorem.

A natural question arises here: What about local connectivity of transcendental maps? For instance we can prove that if $f$ is a transcendental entire map having an unbounded Fatou component $U$, then the Julia set of $f$ cannot be locally connected. On the contrary we claim (the proof is still work in process) that $N_f(z)=z-\tan(z)$ (the Newton's map of $f(z)=\sin(z)$) has infinitely many unbounded compoments (this we know) and the Julia set is locally connected.