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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 ψ_i, the open set condition requires that there exists a nonempty, open set V satisfying two conditions:

$\bigcup _{i=1}^{m}\psi _{i}(V)\subseteq V,$
Each $\psi _{i}(V)$ is 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 measure

When the open set condition holds and each ψ_i is a similitude (that is, a composition of an isometry and a dilation around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for s of the following:^[5]

\sum _{i=1}^{m}r_{i}^{s}=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 ψ_i be the dilation of ratio 1/2 around a_i. The unique non-empty fixed point of the corresponding mapping ψ 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.^[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.

References

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

[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". Proceedings-Cambridge Philosophical Society. 42: 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).

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@@ Line 1: / Line 1: @@
 {{Short description|Condition for self-similar fractals}}
 [[File:Open set condition.png|thumb|an open set covering of the sierpinski triangle along with one of its mappings ψ<sub>''i''</sub>.]]
-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.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ<sub>''i''</sub>, the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
+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.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ<sub>''i''</sub>, the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
 #<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
 # Each <math>\psi_i (V)</math> is pairwise disjoint.
@@ Line 22: / Line 22: @@
 Taking [[natural logarithm]]s 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.<ref>{{Cite web | url=http://www.stat.uchicago.edu/~lalley/Papers/packing.pdf| title=The Packing and Covering Functions for Some Self-similar Fractals|last=Lalley|first=Steven|publisher=Purdue University|date=21 January 1988|access-date=2 February 2022}}</ref> 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.<ref>{{Cite web| url=http://users.jyu.fi/~antakae/publications/preprints/009-controlled_moran.pdf| title=Separation Conditions on Controlled Moran Constructions| last1=Käenmäki| first1=Antti| last2=Vilppolainen| first2=Markku| access-date = 2 February 2022}}</ref><ref>{{Cite journal| last=Schief| first=Andreas| title=Self-similar Sets in Complete Metric Spaces| journal=Proceedings of the American Mathematical Society| volume=124| issue=2| year=1996| url=https://www.ams.org/journals/proc/1996-124-02/S0002-9939-96-03158-9/S0002-9939-96-03158-9.pdf}}</ref> In these cases, SOCS is indeed a stronger condition.
 ==See also==