Attractor sets and Julia sets in low dimensions

TL;DR

This paper establishes a connection between attractor sets of conformal iterated function systems in low dimensions and Julia sets of quasiregular semigroups, showing they can be quasiconformally equivalent under certain conditions.

Contribution

It proves that attractor sets of conformal IFS in dimensions two and three are Julia sets of quasiregular semigroups, with a special case for dimension two using a single generator.

Findings

Attractor sets are Julia sets of quasiregular semigroups.

In dimension two, attractor sets are quasiconformally equivalent to Julia sets of rational maps.

The results extend to semigroups generated by one element under additional conditions.

Abstract

If $X$ is the attractor set of a conformal IFS in dimension two or three, we prove that there exists a quasiregular semigroup $G$ with Julia set equal to $X$. We also show that in dimension two, with a further assumption similar to the open set condition, the same result can be achieved with a semigroup generated by one element. Consequently, in this case the attractor set is quasiconformally equivalent to the Julia set of a rational map.