Pure unrectifiability of fractal sets in the plane

Abstract

A set in the plane is purely unrectifiable if it intersects every curve of finite length, in a set of zero 1-dimensional Hausdorff measure. This dissertation is devoted to investigating the pure unrectifiability of certain classes of fractal sets in the plane, specifically those with infinite 1-dimensional Hausdorff measure. For Jordan curves in the plane, we study the relationship between the set of (inner) tangent points and pure unrectifiability. For quasicircles, we show that there is no relationship between its Hausdorff dimension and the quasicircle being purely unrectifiable. For a Jordan curve, we study the relationship between the 1-dimensional Hausdorff measure and harmonic measure with respect to one of the complementary components. This allows us to obtain a sufficient condition for the pure unrectifiability of a Jordan curve for which one of the complementary components is a one sided quasidisk, namely it is quasiconvex. Using these results, we give examples of purely unrectifiable Jordan curve Julia sets. In particular, we show that the Julia set of $z\mapsto z^2+1/4$ is purely unrectifiable and conclude by giving two applications of this work. We ask the question to classify all non purely unrectifiable quadratic Julia sets. In doing so, we survey various results on Julia sets containing rectifiable arcs and argue for the need to investigate the finer structure of quadratic Julia sets via pure unrectifiability. Lastly, we discuss some open questions we intend to pursue in the near future.