Combinatorial rigidity for some infinitely renormalizable unicritical polynomials

Abstract

Abstract. Here we prove that infinitely renormalizable unicritical polynomials P c : z → z d + c, with c ∈ C, satisfying a priori bounds and a certain "combinatorial" condition are combinatorially rigid. This implies the local connectivity of the connectedness loci (the Mandelbrot set when d = 2) at the corresponding parameters.

A survey on MLC, Rigidity and related topics

“…A quite standard technique which is used to show rigidity is to construct quasiconformal conjugacies between combinatorially equivalent maps by using a pull-back argument, and then using an open-closed argument (see Section 8 and [Che10]) to deduce that combinatorial classes consist of a single point. The Pullback Argument was probably introduced by Thurston for postcritically finite maps, and has been used several times since then (see among others [Su88, Paragraph 11], [Lyu97] and [Che10], [KSvS07] for polynomials, [Cui01] for rational maps, [Be15] for exponential maps. In order to be able to take pullbacks, one needs some information on the behaviour of the postsingular set-ideally, the singular value is non-recurrent, but one can also deal with weak forms of recurrence.…”

“…Several results are known about absence of line fields for different classes of maps, one can find a summary in the paper [YZ10], where the authors use puzzle techniques to show that a rational map carries no invariant line field when its Julia set is a Cantor set. Another important result is a rigidity theorem by McMullen [McM94, Theorem 10.2] stating that there are no invariant line fields for infinitely renormalizable quadratic polynomial-like maps which satisfy a priori bounds (the proof also works for degree d unicritical polynomial maps [Che10]). Additional results for absence of invariant line fields for rational maps, obtained using quadratic differentials, can be found in [Ma05].…”

“…Complex bounds have been introduced in [Su88] and have been widely used thereafter (see [Lev11] for a list of works using complex bounds). Roughly speaking, a priori bounds imply that a sequence of renormalizations is precompact, allowing the usage of a well established machinery based on Sullivan-Thurston pullback argument to obtain combinatorial rigidity (see for example [Che10]). So in many cases, the main difficulty when studying a new combinatorial class of (infinitely renormalizable) parameters relies consists in proving some kind of a priori bounds.…”

“…Remark 8.1. In [Lyu97], [Che10], the combinatorics of an infinitely renormalizable polynomial P c is defined as the sequence of maximal Mandelbrot copies which contain the parameters corresponding to the successive renormalization of c. This is indeed the same concept. One can translate the definition in [Lyu97] to obtain a sequence of nested maximal Mandelbot copies which actually contain c, and each of which corresponds to a renormalization.…”

“…Some of this material naturally overlaps with previous writings, especially with [McM94] and [Hu93]. However, we include: recent rigidity results by Levin for classes of maps with no a priori bounds ( [Lev11], [Lev14]); a section on parabolic and near parabolic renormalization for quadratic polynomials ( [IS06], [CS15]); results by Avila, Kahn, Lyubich and Shen and by Cheraghi about rigidity and local connectivity for unicritical polynomials ([AKLS09], [KL09b], [Che10]); and some recent results on topological models [BOPT16]. We also present a description of the combinatorial approach to the MLC conjecture which is now widely used ( [Sc04], [RS08]) and a unified view of the relations between the main conjectures related to MLC.…”