2026/07/31 by Neil Dobbs, Jacek Graczyk, Nicolae Mihalache
Mathematics · #math.DS #msc:37F10 #msc:37F35 #msc:37F40 #msc:37F44 #msc:37A10 #msc:37E05
39 pages, 2 figures
arxiv created 2026/07/31 · arxiv updated 2026/08/03
For the quadratic family fc(z)=z2+c, the only parameters in the Mandelbrot set \cal M for which the Julia set \cal Jc 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 \cal M, however, is a non-hyperbolic parameter and the dimension function c↦ dimH(\cal Jc) is highly discontinuous there. We prove the sharp first-order asymptotic for the lower envelope of the Hausdorff dimension at the tip: If c∈ \cal M then dimH(\cal Jc) lies asymptotically above 1+ Ω√(|c+2|) with the Jaksztas constant Ω=√((2)/(3))(1)/(πlog 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→ -2, the induced dynamics exhibit a uniform property of exponential tails, generating improved control of their pressure functions.