In fractal geometry, the open set condition (OSC) is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction. Specifically, given an iterated function system of contractive mappings ψ 1 , … , ψ m {\displaystyle \psi _{1},\ldots ,\psi _{m}} , the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
⋃ i = 1 m ψ i ( V ) ⊆ V , {\displaystyle \bigcup _{i=1}^{m}\psi _{i}(V)\subseteq V,}
The sets ψ 1 ( V ) , … , ψ m ( V ) {\displaystyle \psi _{1}(V),\ldots ,\psi _{m}(V)} are pairwise disjoint. Introduced in 1946 by P.A.P Moran, the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure. An equivalent statement of the open set condition is to require that the s-dimensional Hausdorff measure of the set is greater than zero.
Computing Hausdorff dimension When the open set condition holds and each ψ i {\displaystyle \psi _{i}} is a similitude (that is, a composition of an isometry and a dilation around some point), then the unique fixed point of ψ {\displaystyle \psi } is a set whose Hausdorff dimension is the unique solution for s of the following:
∑ i = 1 m r i s = 1. {\displaystyle \sum _{i=1}^{m}r_{i}^{s}=1.}
where ri is the magnitude of the dilation of the similitude. With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three non-collinear points a1, a2, a3 in the plane R2 and let ψ i {\displaystyle \psi _{i}} be the dilation of ratio 1/2 around ai. The unique non-empty fixed point of the corresponding mapping ψ {\displaystyle \psi } is a Sierpinski gasket, and the dimension s is the unique solution of
( 1 2 ) s + ( 1 2 ) s + ( 1 2 ) s = 3 ( 1 2 ) s = 1. {\displaystyle \left({\frac {1}{2}}\right)^{s}+\left({\frac {1}{2}}\right)^{s}+\left({\frac {1}{2}}\right)^{s}=3\left({\frac {1}{2}}\right)^{s}=1.}
Taking natural logarithms of both sides of the above equation, we can solve for s, that is: s = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.
Strong open set condition The strong open set condition (SOSC) is an extension of the open set condition. A fractal F satisfies the SOSC if, in addition to satisfying the OSC, the intersection between F and the open set V is nonempty. The two conditions are equivalent for self-similar and self-conformal sets, but not for certain classes of other sets, such as function systems with infinite mappings and in non-euclidean metric spaces. In these cases, SOCS is indeed a stronger condition.
See also Cantor set List of fractals by Hausdorff dimension Minkowski–Bouligand dimension Packing dimension
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