On the size of Siegel disks with fixed multiplier for cubic polynomials

Abstract: We study the slices of the parameter space of cubic polynomials where we fix the multiplier of a fixed point to some value $\lambda$. The main object of interest here is the radius of convergence of the linearizing parametrization. The opposite of its logarithm turns out to be a sub-harmonic function of the parameter whose Laplacian $\mu_\lambda$ is of particular interest. We relate its support to the Zakeri curve in the case the multiplier is neutral with a bounded type irrational rotation number. In the attracting case, we define and study an analogue of the Zakeri curve, using work of Petersen and Tan. In the parabolic case, we define an analogue using the notion of asymptotic size. We prove a convergence theorem of $\mu_{\lambda_n}$ to $\mu_\lambda$ for $\lambda_n= \exp(2\pi i p_n/q_nn)$ and $\lambda = \exp(2\pi i\theta)$ where $\theta$ is a bounded type irrational and $p_n/q_n$ are its convergents.