Dimension paradox of irrationally indifferent attractors
arXiv:2003.12340
Abstract
We prove that for an infinite dimensional class of holomorphic maps with an elliptic fixed point, the post-critical set has Hausdorff dimension two, provided the rotation number is non-Herman of sufficiently high type. We identify classes of Brjuno but not Herman, and non-Brjuno, numbers such that the post-critical set satisfies the Karpinska's dimension paradox. That is, the set of end points of the post-critical set has dimension two, but without those end points, the dimension drops to one.
32 pages, 4 figures, v2: substantial rewriting of the first draft -- improving the presentation with simplifying and clarifying many arguments