Self-similarity and Julia Sets

The symmetry of a measure space \((X, \mu )\) is described by measure-preserving group actions it admits. Such group actions naturally give rise to a unitary representation, called the Koopman representation, of the group on \(L^2(X, \mu )\) . On the other hand, the symmetry of \((X, \mu )\) also enables the construction of groups with certain desired properties. Of particular interest is the case when \(X=[0, 1]\) with Lebesgue measure \(\mu \) , through which the first example of a group of intermediate growth, the Grigorchuk group \(\mathcal {G}\) , was constructed [103]. Subsequently, it was discovered that \(\mathcal {G}\) displays a self-similarity property.