Attractors of sequences of iterated function systems

TL;DR

This paper investigates whether complex limit sets generated by sequences of different iterated function systems can be represented by a single system, revealing limitations under certain conditions relevant to quasicrystal spectral theory.

Contribution

It proves that, under specific assumptions, certain Cantor sets from sequences of IFS cannot be realized by a single smooth IFS, highlighting fundamental constraints in the structure of these fractals.

Findings

Certain Cantor sets cannot be generated by a single $C^{1 + eta}$ IFS under specified conditions.

The results have implications for the spectral theory of one-dimensional quasicrystals.

The paper establishes conditions where the limit sets of iterated function systems are not equivalent to those of a single system.

Abstract

If $F$ and $G$ are iterated function systems, then any infinite word $W$ in the symbols $F$ and $G$ induces a limit set. It is natural to ask whether this Cantor set can also be realized as the limit set of a single $C^{1 + \alpha}$ iterated function system $H$. We prove that under certain assumptions on $F$, $G$ and $H$, the answer is no. This problem is motivated by the spectral theory of one-dimensional quasicrystals.