Distortion and topology

Abstract

For a self mapping $f:\mathbb{D}\to \mathbb{D}$ of the unit disk in $\mathbb{C}$ which has finite distortion, we give a separation condition on the components of the set where the distortion is large - say greater than a given constant - which implies that $f$ extends homeomorphically and quasisymetrically to the boundary $\mathbb{S}$ and thus $f$ shares its boundary values with a quasiconformal mapping whose distortion can be explicitly estimated in terms of the data. This result holds more generally. This condition, uniformly separated in modulus, allows the set where the distortion is large to accumulate densely on the boundary but does not allow a component to run out to the boundary. The lift of a Jordan domain in a Riemann surface to its universal cover $\mathbb{D}$ is always uniformly separated in modulus and this allows us to apply these results in the theory of Riemann surfaces to identify an interesting link between the support of the high distortion of a map and topology of the surface - again with explicit and good estimates. As part of our investigations we study mappings $φ:\mathbb{S}\to\mathbb{S}$ which are the germs of a conformal mapping and give good bounds on the distortion of a quasiconformal extension of $φ$. We extend these results to the germs of quasisymmetric mappings. These appear of independent interest and identify new geometric invariants.