Topologie locale des méthodes de Newton cubiques: plan dynamique

Abstract

For a cubic Newton map N, we obtain the following theorems: 1) The boundary of the immediate basin of each fixed critical point is locally connected. 2) The Julia set J(N) is locally connected provided either N has no irrational indifferent periodic point or N has no Siegel disc and the orbit of the non-fixed critical point doesn 't accumulate on the boundary of the fixed immediate basins. In particular, in contrast with Julia sets of polynomials, J(N) can be locally connected even if N has a periodic Cremer point. The proofs rely on the construction of articulated rays which are very special simple arcs landing on J(N).