On the Hausdorff dimension of the Gieseking fractal

Let φ :R→C be the Cannon–Thurston map associated to the Gieseking manifold; thus φ is also the Cannon–Thurston map associated to the complement of the figure-eight knot. There are sets ER and EC such that φ decomposes into two maps ER→EC and R−ER→C−EC, where the former is (at-least-two)-to-one, while the latter is a homeomorphism. It is known that ER has Hausdorff dimension zero. Let d denote the Hausdorff dimension of EC. We show that EC is a countable union of Jordan curves of Hausdorff dimension d, and that d=1.35±0.2. In particular, the two-dimensional Lebesgue measure of EC is zero, and the Jordan curves are not rectifiable. We also show that d is the critical exponent of a Poincaré series of a discrete semigroup of isometries of hyperbolic three-space, and describe a computer experiment that suggests that d is near 1.2971.