# Open set condition

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{{Short description|Condition for fractals in math}}
[[File:Open set condition.png|thumb|an open set covering of the [sierpinski triangle](/source/sierpinski_triangle) along with one of its mappings ψ<sub>''i''</sub>.]]
In [fractal geometry](/source/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](/source/iterated_function_system) of [contractive mappings](/source/contraction_mapping) <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](/source/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](/source/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](/source/isometry) and a [dilation](/source/dilation_(metric_space)) around some point), then the unique fixed point of <math>\psi</math> is a set whose [Hausdorff dimension](/source/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](/source/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 logarithm](/source/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==
*[Cantor set](/source/Cantor_set)
*[List of fractals by Hausdorff dimension](/source/List_of_fractals_by_Hausdorff_dimension)
*[Minkowski–Bouligand dimension](/source/Minkowski%E2%80%93Bouligand_dimension)
*[Packing dimension](/source/Packing_dimension)

==References==
{{reflist}}

Category:Iterated function system fractals

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