Open set condition


In fractal geometry, the open set condition 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, the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
  1. The sets 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 is a similitude, then the unique fixed point of is a set whose Hausdorff dimension is the unique solution for s of the following:
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 be the dilation of ratio 1/2 around ai. The unique non-empty fixed point of the corresponding mapping is a Sierpinski gasket, and the dimension s is the unique solution of
Taking natural logarithms of both sides of the above equation, we can solve for s, that is: s = ln/ln. The Sierpinski gasket is self-similar and satisfies the OSC.

Strong open set condition

The strong open set condition 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.