Iterated function systems with super-exponentially close cylinders II

TL;DR

This paper proves a general theorem on the existence of iterated function systems with super-exponentially close cylinders at all small scales, expanding understanding of fractal geometry and overlaps.

Contribution

It introduces a new theorem showing such systems exist within parameterized families under certain conditions, with explicit examples provided.

Findings

Existence of IFS with no exact overlaps and super-exponentially close cylinders.

A general theorem linking parameterized families to the existence of such IFS.

Explicit examples demonstrating the theorem's applicability.

Abstract

Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small scales. Iterated function systems satisfying these properties were shown to exist by the author and by B\'{a}r\'{a}ny and K\"{a}enm\"{a}ki. In this paper we prove a general theorem on the existence of such iterated function systems within a parameterised family. This theorem shows that if a parameterised family contains two independent subfamilies, and the set of parameters that cause exact overlaps satisfies some weak topological assumptions, then the original family will contain an iterated function system satisfying the desired properties. We include several explicit examples of parameterised families to which this theorem can be applied.