An escape time Sierpiński map is a rational map drawn from the McMullen family z ↦ zⁿ + λ/zⁿ with escaping critical orbits and Julia set homeomorphic to the Sierpiński curve continuum. We address the problem of characterizing postcritically finite escape time Sierpiński maps in a combinatorial way. To accomplish this, we define a combinatorial model given by a planar tree whose vertices come with a pair of combinatorial data that encodes the dynamics of critical orbits. We show...

For the family of rational maps zⁿ + λ/zⁿ where n ≥ 3, it is known that there are infinitely many small copies of the Mandelbrot set that are buried in the parameter plane, i.e., they do not extend to the outer boundary of this set. For parameters lying in the main cardioids of these Mandelbrot sets, the corresponding Julia sets are always Sierpiński curves, and so they are all homeomorphic to one another. However, it is known that only those cardioids that are symmetrically located in the parameter...

We give a new proof of the following conjecture of Yoccoz:$$(\exists C\in ℝ)(\forall \theta \in ℝ\setminus \mathbb{Q})log\mathrm{rad}\Delta \left(Q_{\theta }\right)\le -Y\left(\theta \right)+C,$$where $Q_{\theta }\left(z\right)={\mathrm{e}}^{2\pi i\theta }z+z^2$, $\Delta \left(Q_{\theta }\right)$ is its Siegel disk if $Q_{\theta }$ is linearizable (or $\varnothing$ otherwise), $\mathrm{rad}\Delta \left(Q_{\theta }\right)$ is the conformal radius of the Siegel disk of $Q_{\theta }$ (or $0$ if there is none) and $Y\left(\theta \right)$ is Yoccoz’s Brjuno function.In a former article we obtained a first proof based on the control of parabolic explosion. Here, we present a more elementary proof based on Yoccoz’s initial methods.We then extend this result to some new families of polynomials such as $z^d+c$ with $d\>2$. We also show that...

In a recent preprint [B], Bergweiler relates the number of critical points contained in the immediate basin of a multiple fixed point β of a rational map f: ℙ¹ → ℙ¹, the number N of attracting petals and the residue ι(f,β) of the 1-form dz/(z-f(z)) at β. In this article, we present a different approach to the same problem, which we were developing independently at the same time. We apply our method to answer a question raised by Bergweiler. In particular, we prove that when there are only...

A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we provea priori bounds. They imply local connectivity of the corresponding Julia sets...

We first introduce the class of quasi-algebraically stable meromorphic maps of Pk. This class is strictly larger than that of algebraically stable meromorphic self-maps of Pk. Then we prove that all maps in the new class enjoy a recurrent property. In particular, the algebraic degrees for iterates of these maps can be computed and their first dynamical degrees are always algebraic integers.