{{Short description|Condition for fractals in math}} [[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 contractive mappings <math>\psi_1, \ldots, \psi_m</math>, 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> # The sets <math>\psi_1(V), \ldots, \psi_m(V)</math> are pairwise disjoint.
Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last1=Moran | first1=P. A. P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Mathematical Proceedings of the Cambridge Philosophical Society | volume=42 | issue=1 | year=1946 | pages=15-23 | doi=10.1017/S0305004100022684}}</ref> 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.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>
An equivalent statement of the open set condition is to require that the s-dimensional Hausdorff measure of the set is greater than zero.<ref> {{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>
==Computing Hausdorff dimension== When the open set condition holds and each <math>\psi_i</math> is a similitude (that is, a composition of an isometry and a dilation around some point), then the unique fixed point of <math>\psi</math> is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>
:<math> \sum_{i=1}^m r_i^s = 1. </math>
where r<sub>i</sub> 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 ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub> in the plane '''R'''<sup>2</sup> and let <math>\psi_i</math> be the dilation of ratio 1/2 around ''a<sub>i</sub>''. The unique non-empty fixed point of the corresponding mapping <math>\psi</math> is a Sierpinski gasket, and the dimension ''s'' is the unique solution of :<math> \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. </math>
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.<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== *Cantor set *List of fractals by Hausdorff dimension *Minkowski–Bouligand dimension *Packing dimension
==References== {{reflist}}
Category:Iterated function system fractals