Open set condition explained

\psi _{1,} \ldots, \psi _{m} , the open set condition requires that there exists a nonempty, open set V satisfying two conditions: c up ^{m\psi} _{i} (V) \subseteq V,

1. The sets

\psi _{1(V),} \ldots, \psi _{m(V)} are pairwise disjoint.

Introduced in 1946 by P.A.P Moran,^[2] 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.^[3]

An equivalent statement of the open set condition is to require that the s-dimensional Hausdorff measure of the set is greater than zero.^[4]

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:^[5] \sum ^{m} _{i =1} r ^{s} _{i} = 1 .

where r_i 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 a_i. 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( (1)/(2) \right) ^{s + \left( (1)/(2)} \right) ^{s + \left( (1)/(2)} \right) ^{s} = 3 \left( (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.^[6] 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.^{[7] [8]} In these cases, SOCS is indeed a stronger condition.

See also

• Cantor set
• List of fractals by Hausdorff dimension
• Minkowski–Bouligand dimension
• Packing dimension

Notes and References

1. Bandt . Christoph . Viet Hung . Nguyen . Rao . Hui . On the Open Set Condition for Self-Similar Fractals . Proceedings of the American Mathematical Society . 134 . 2006 . 1369–74 . 5 . limited.
2. Moran . P.A.P. . Additive Functions of Intervals and Hausdorff Measure . Proceedings of the Cambridge Philosophical Society . 42 . 1946 . 15-23 . 10.1017/S0305004100022684.
3. Llorente. Marta. Mera. M. Eugenia. Moran. Manuel. On the Packing Measure of the Sierpinski Gasket . University of Madrid .
4. Web site: Open set condition for self-similar structure . Wen . Zhi-ying . Tsinghua University . 1 February 2022 .
5. Hutchinson . John E. . Fractals and self similarity . Indiana Univ. Math. J. . 30 . 1981 . 713–747 . 10.1512/iumj.1981.30.30055 . 5 . free .
6. Web site: The Packing and Covering Functions for Some Self-similar Fractals. Lalley. Steven. Purdue University. 21 January 1988. 2 February 2022.
7. Web site: Separation Conditions on Controlled Moran Constructions. Käenmäki. Antti. Vilppolainen. Markku. 2 February 2022.
8. Schief. Andreas. Self-similar Sets in Complete Metric Spaces. Proceedings of the American Mathematical Society. 124. 2. 1996.