Open set condition

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 $$\psi_1, \ldots, \psi_m$$, the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
 * 1) $$ \bigcup_{i=1}^m\psi_i (V) \subseteq V, $$
 * 2) The sets $$\psi_1(V), \ldots, \psi_m(V)$$ 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 $$\psi_i$$ is a similitude (that is, a composition of an isometry and a dilation around some point), then the unique fixed point of $$\psi$$ is a set whose Hausdorff dimension is the unique solution for s of the following:


 * $$ \sum_{i=1}^m r_i^s = 1. $$

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 $$\psi_i$$ be the dilation of ratio 1/2 around ai. The unique non-empty fixed point of the corresponding mapping $$\psi$$ is a Sierpinski gasket, and the dimension s is the unique solution of
 * $$ \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. $$

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.