On the computability of fractal dimensions and Hausdorff measure

Abstract

It is shown that there exist subsets A and B of the real line which are recursively constructible such that A has a nonrecursive Hausdorff dimension and B has a recursive Hausdorff dimension (between 0 and 1) but has a finite, nonrecursive Hausdorff measure. It is also shown that there exists a polynomial-time computable curve on the two-dimensional plane that has a nonrecursive Hausdorff dimension between 1 and 2. Computability of Julia sets of computable functions on the real line is investigated. It is shown that there exists a polynomial-time computable function f on the real line whose Julia set is not recurisvely approximable.