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After giving an introduction to the procedure dubbed and quickly recalling known results about more classical notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree $3$ post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree $6$ rational map. We give a conjectural interpretation...

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...