Ghys-like models providing trick for a class of simple maps

Abstract

For quadratic polynomials with an indifferent fixed point with bounded type rotation number (they have a Siegel disk), much of what is known of their Julia set comes from the study of a quasiconformal model. The model is build from a Blaschke fraction, that we call a pre-model, and that is given by a formula. We give here a geometric construction of pre-model maps, that extends to some cases where no formula is known. More precisely, we are able to make this work for a class of entire maps, very specific but nonetheless spanning uncountably many equivalence classes (thus with probably no hope for a formula), and also in the case of the Lavaurs maps that arise in the parabolic implosion of quadratic polynomials.