Asymptotic variance of the Beurling transform

We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a $k$-quasicircle is at most $1+k^2$, it is natural to expect that the maximum asymptotic variance $\Sigma^2 = 1$. In this paper, we prove $0.87913 \le \Sigma^2 \le 1$. For the lower bound, we give examples of polynomial Julia sets which are $k$-quasicircles with dimensions $1+ 0.87913 \, k^2$ for $k$ small, thereby showing that $\Sigma^2 \ge 0.87913$. The key ingredient in this construction is a good estimate for the distortion $k$, which is better than the one given by a straightforward use of the $\lambda$-lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating $\Sigma^2$ in terms of nearly circular polynomial Julia sets.