Iterated function system quasiarcs

TL;DR

This paper studies a special class of iterated function systems in Euclidean space, providing necessary conditions for their invariant sets to be parameterized in a quasisymmetric manner when they are homeomorphic to an interval.

Contribution

It establishes necessary conditions based solely on the similarities for invariant sets of IFSs to have quasisymmetric parameterizations when homeomorphic to an interval.

Findings

Identifies conditions for quasisymmetric parameterization of IFS invariant sets.

Provides criteria for when these sets are homeomorphic to an interval.

Connects geometric properties of IFSs with topological and metric parameterizations.

Abstract

We consider a class of iterated function systems (IFSs) of contracting similarities of $R^n$, introduced by Hutchinson, for which the invariant set possesses a natural H\"older continuous parameterization by the unit interval. When such an invariant set is homeomorphic to an interval, we give necessary conditions in terms of the similarities alone for it to possess a quasisymmetric (and as a corollary, bi-H\"older) parameterization. We also give a related necessary condition for the invariant set of such an IFS to be homeomorphic to an interval.