Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets

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Quantization dimension for conformal iterated function systems (2002, background): “An iterated function system satisfies the open set condition if there exists a non-empty open set U ⊂ X such that ϕi(U) ⊂ U for every i ∈ I and ϕi(X) ∩ ϕj (X) =∅ for every pair i, j ∈ I , i �= j . Furthermore, the system satisfies the strong open set condition if U can be chosen such that U ∩ J � =∅”
Developments in fractal geometry (2013, unknown): “Various reﬁnements and modiﬁcations of the open set condition are used to yield additional information about dimensions of attractors; see for example [102] and the references therein.”
Assouad Dimension and Fractal Geometry (2020, background): “The first is that the OSC is equivalent to H(F ) > 0 where P (s) = 0, which is a result of Peres, Rams, Simon and Solomyak [231].”
On natural invariant measures on generalised iterated function systems (2017, background): “Peres, Rams, Simon and Solomyak [ 17 ] generalised the result of Schief for the conformal setting.”