The exact overlap conjecture for one-dimensional iterated function systems

Let $\mathcal{F}$ be an iterated function system with attractor $\Lambda$, alphabet $\mathcal{A}$, and similarity dimension $s^{\mathcal{F}}$, the unique zero of its pressure function. An exact overlap means that there exist distinct finite words $\mathbf{i},\mathbf{j}\in\bigcup_n\mathcal{A}^n$ such that

$$f_{\mathbf{i}}|_{\Lambda}\equiv f_{\mathbf{j}}|_{\Lambda}.$$

Exact overlap conjecture. If

$$\dim_{\rm H}\Lambda<\min\{1,s^{\mathcal{F}}\},$$

then $\mathcal{F}$ has an exact overlap.

The conjecture predicts that a strict drop of Hausdorff dimension below the natural upper bound can occur only through an exact algebraic coincidence between cylinder maps. It extends the familiar separation-condition formula for the dimension and remains unresolved in the generality stated here.