Precise asymptotics at the tip of the Mandelbrot set

Abstract

For the quadratic family $f_c(z)=z^2+c$, the only parameters in the Mandelbrot set $\mathcal{M}$ for which the Julia set ${\mathcal{J}}_c$ has Hausdorff dimension $1$ are $c=0$ and $c=-2$. Near $c=0$, Ruelle's theory gives a real-analytic expansion of the dimension. The tip $c=-2$ of $\mathcal{M}$, however, is a non-hyperbolic parameter and the dimension function $c\mapsto \mathrm{di}{\mathrm{m}}_{\mathrm{H}}({\mathcal{J}}_c)$ is highly discontinuous there. We prove the sharp first-order asymptotic for the lower envelope of the Hausdorff dimension at the tip: If $c\in \mathcal{M}$ then $\mathrm{di}{\mathrm{m}}_{\mathrm{H}}({\mathcal{J}}_c)$ lies asymptotically above $1+\Omega \sqrt{|c+2|}$ with the Jaksztas constant $\Omega =\sqrt{\frac{2}{3}}\frac{1}{\pi \log 2}$. This is a surprisingly precise contribution to the Yoccoz problem about unfolding attractors. The proof develops a thermodynamic formalism for degenerating families of box mappings. At each scale, for parameters $c\to -2$, the induced dynamics exhibit a uniform property of exponential tails, generating improved control of their pressure functions.

39 pages, 2 figures