Editing Hausdorff dimension (section)

===The open set condition===
{{main|Open set condition}}
To determine the dimension of the self-similar set ''A'' (in certain cases), we need a technical condition called the ''open set condition'' (OSC) on the sequence of contractions ψ<sub>''i''</sub>.

There is an open set ''V'' with compact closure, such that

:<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>

where the sets in union on the left are pairwise [[disjoint sets|disjoint]].

The open set condition is a separation condition that ensures the images ψ<sub>''i''</sub>(''V'') do not overlap "too much".

'''Theorem'''. Suppose the open set condition holds and each ψ<sub>''i''</sub> is a similitude, that is a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is ''s'' where ''s'' is the unique solution of<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>

The contraction coefficient of a similitude is the magnitude of the dilation.

In general, a set ''E'' which is carried onto itself by a mapping

: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>

is self-similar if and only if the intersections satisfy the following condition:

:<math> H^s\left(\psi_i(E)\cap \psi_j(E)\right) =0, </math>

where ''s'' is the Hausdorff dimension of ''E'' and ''H<sup>s</sup>'' denotes s-dimensional [[Hausdorff measure]]. This is clear in the case of the [[Sierpinski gasket]] (the intersections are just points), but is also true more generally:

'''Theorem'''. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.