Quasicircles and hyperbolic zero packing

TL;DR

This paper explores the properties of quasicircles and hyperbolic zero packing, focusing on the integral means spectrum of conformal maps with small quasiconformal dilatation, making Ivrii's work more accessible.

Contribution

It provides an accessible exposition of Ivrii's work on quasicircle dimensions and analyzes the integral means spectrum for conformal maps with small quasiconformal extensions.

Findings

Upper estimates for the integral means spectrum are established.

Connections between quasicircle dimensions and hyperbolic zero packing are clarified.

The work simplifies understanding of Ivrii's results on small quasiconformality.

Abstract

We look at the work of Oleg Ivrii connected with the dimension of quasicircles for asymptotically small quasiconformality parameter $k$. We intend to make this work more easily accessible. Our main focus is the integral means spectrum associated with normalized conformal mappings of the exterior disk which have quasiconformal extensions to the whole plane with small dilatation parameter $k$. Moreover, we address the estimates from above only, not the sharpness from below.