open set condition

File:Open set condition.png along with one of its mappings ψ_i.]]

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.{{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}} Specifically, given an iterated function system of contractive mappings $\psi_1, \ldots, \psi_m$ , the open set condition requires that there exists a nonempty, open set V satisfying two conditions:

$\bigcup_{i=1}^m\psi_i (V) \subseteq V,$
The sets $\psi_1(V), \ldots, \psi_m(V)$ are pairwise disjoint.

Introduced in 1946 by P.A.P Moran,{{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}} 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.{{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}}

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

{{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 }}

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:{{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 }}

: $\sum_{i=1}^m r_i^s = 1.$

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 $\psi_i$ 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

: $\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1.$

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.{{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}} 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.{{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}}{{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}} In these cases, SOCS is indeed a stronger condition.

References

Category:Iterated function system fractals

open set condition

Computing Hausdorff dimension

Strong open set condition

See also

References