Slices of the parameter space of cubic polynomials

Abstract

We construct a locally connected model of the boundary of the cubic connectedness locus. The model is obtained by constructing a decomposition of the space of critical portraits and collapsing elements of the decomposition into points. This model is similar to a quotient of the quadratic combinatorial locus where all baby Mandelbrot sets are collapsed to points. All fibers of the model, possibly except one, are connected.

Lavaurs algorithm for cubic symmetric polynomials

“…A natural next object of study is M 3 , and some slices of M 3 have already been considered. In [BOSTV1] we construct the lamination C s CL (this stands for cubic symmetric comajor lamination) together with the induced factor space S/C s CL of the unit circle S. In [BOSTV3] we verify that S/C s CL is a monotone model of the cubic symmetric connected locus, i.e. the space M 3,s of symmetric cubic polynomials P (z) = z 3 + λ 2 z with connected Julia sets.…”

Upper bounds for the moduli of polynomial-like maps

“…On the other hand, the structure of the neutral slices F λ with |λ| = 1 is much more delicate. There are still many unanswered questions about them (cf [11]); see [10,48,49] for very recent results concerning the simplest neutral slices(where λ = e 2π iθ , with θ being rational or irrational of bounded type). An earlier paper [44] studies the dynamics of polynomials from F 1 .…”

“…Theorem 2.10 verifies a conjecture from [5]; it is the main result of this paper concerning polynomial parameter spaces. By the Main Theorem of [11] (see theorem 5.14), the set CU λ is a full continuum.…”

“…Lemma 2.11 (Lemma 7.2 [11]). If f is a complex cubic polynomial with a non-repelling fixed point a, and there exists a quadratic-like filled Julia set K * with a ∈ K * , then K * is unique.…”

“…By theorem 2.10, we have P / ∈ CU λ . By the main theorem of [11] (see theorem 5.14), the polynomial P belongs to a wake W λ (θ 1 , θ 2 ), and the rays R P (θ 1 + 1/3), R P (θ 2 + 2/3) land at a repelling or parabolic point a; in the latter case a = 0. These rays and a form a cut whose cycle is the desired Z because the wedge formed by the rays R P (θ 1 + 1/3), R P (θ 2 + 2/3) has the angle measure greater than 1/3 and, hence, must contain ω 2 (P).…”