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Open set condition

an open set covering of the sierpinski triangle along with one of its mappings ψi.

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.[1] 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:

  2. The sets 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 is a similitude (that is, a composition of an isometry and a dilation around some point), then the unique fixed point of is a set whose Hausdorff dimension is the unique solution for s of the following:[5]

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(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

References

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