Iterated function systems with super-exponentially close cylinders

TL;DR

This paper constructs examples of iterated function systems in real numbers that lack exact overlaps but still have cylinders that are super-exponentially close at all small scales, challenging previous conjectures.

Contribution

It proves the existence of non-overlapping IFS with super-exponentially close cylinders at all scales, extending understanding of fractal measure dimensions.

Findings

Existence of IFS without exact overlaps but with super-exponentially close cylinders

Construction of IFS matching any prescribed sequence of closeness thresholds

Implications for conjectures relating overlaps and measure dimensions

Abstract

Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then the underlying iterated function system contains an exact overlap. In recent years significant progress has been made towards these conjectures. Hochman proved that if the Hausdorff dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then there are cylinders which are super-exponentially close at all small scales. Several years later, Shmerkin proved an analogous statement for the $L^q$ dimension of self-similar measures in $\mathbb{R}$. With these statements in mind, it is natural to wonder whether there exist iterated function systems that do not contain exact overlaps, yet there are cylinders which are super-exponentially close at all small scales. In this paper we show that such iterated function systems do exist. In fact we prove much more. We prove that for any sequence $(\epsilon_n)_{n=1}^{\infty}$ of positive real numbers, there exists an iterated function system $\{\phi_i\}_{i\in \mathcal{I}}$ that does not contain exact overlaps and $$\min\left\{|\phi_{\mathbf{a}}(0)-\phi_{\mathbf{b}}(0)|: \mathbf{a},\mathbf{b}\in \mathcal{I}^n,\, \mathbf{a}\neq \mathbf{b},\, r_{\mathbf{a}}=r_{\mathbf{b}}\right\}\leq \epsilon_n$$ for all $n\in \mathbb{N}.$