Let d≥ 1 and let Phi=phi_i_i=1^m be a finite iterated function system of similitudes on R^d with non-singleton attractor K. Suppose that Phi satisfies the open set condition but not the strong separation condition. Is it possible for there to exist another finite iterated function system of similitudes Psi on R^d whose attractor is the same set K and which satisfies the strong separation condition? Equivalently, must every finite similitude presentation of K fail the strong separation condition whenever one such presentation satisfies the open set condition but not the strong separation condition?

Separation-conditions folklore question for self-similar sets

Let $d\ge 1$ and let $\Phi=\{\phi_i\}_{i=1}^m$ be a finite iterated function system of similitudes on $\mathbb{R}^d$ with non-singleton attractor $K$ . Suppose that $\Phi$ satisfies the open set condition but not the strong separation condition. Is it possible for there to exist another finite iterated function system of similitudes $\Psi$ on $\mathbb{R}^d$ whose attractor is the same set $K$ and which satisfies the strong separation condition? Equivalently, must every finite similitude presentation of $K$ fail the strong separation condition whenever one such presentation satisfies the open set condition but not the strong separation condition?

References

Primary source

arXiv

Additional references

Separation Conditions for Iterated Function Systems with a Common Attractor — arXiv — Cai-Yun Ma

dynamical-systems fractal-geometry

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims the folklore question has a negative answer in all finite Euclidean cases, but that claim has not been independently verified.

The question asks whether an attractor generated with the open separation condition but not the strong separation condition can have another finite similitude presentation with the strong separation condition. The earlier obstruction required equal contraction ratios; the new claim removes that restriction.

Known results

In 2022, an obstruction theorem treated homogeneous systems satisfying $\mathrm{OSC}$ but not $\mathrm{SSC}$ , proving that their attractor cannot arise from any finite similitude system satisfying $\mathrm{SSC}$ .

September 15, 2026 claimed resolution

On September 15, 2026, Cai-Yun Ma's preprint Separation Conditions for Iterated Function Systems with a Common Attractor claimed to remove homogeneity from the obstruction theorem, giving a negative answer to the folklore question. This would settle the stated problem, but the claim is unverified.

Current status (as of September 2026): The homogeneous case is established, while the claimed general negative answer remains independently unverified.

Sources

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