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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 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
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 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
'''定理'''假设开集条件成立，并且每个ψ<sub>''i''</sub> 是一个相似量，这是一个等距和围绕某一点的膨胀的合成。那么唯一的不动点是一个集合，它的豪斯多夫维数是 ''s'' ，其中 ''s'' 是 ''s'' <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>的唯一解
'''定理'''假设开集条件成立，并且每个ψ<sub>''i''</sub> 是一个相似度，即等距和某个点周围的膨胀的组合。。那么唯一的不动点是Hausdorff维数为 ''s'' 的集合，其中 ''s'' 是 ''s'' <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>的唯一解
:<math> \sum_{i=1}^m r_i^s = 1. </math>
:<math> \sum_{i=1}^m r_i^s = 1. </math>