Self-similar groups in the sense of iterated function system

Self-similarity is the most important property of the classical fractals. Self-similarity of the automorphism group of a rooted tree is defined by S. Sidki (Definition 3.1.3). The well-known examples of this group are Adding machine, Grigorchuk and infinite dihedral group. On the other hand, the notion of self-similarity in the sense of iterated function system (IFS) for compact topological groups is given by Ş. Koçak (Definition 4.1.1).In this work, first we investigate self-similar group structure in the sense of Definition 3.1.3. We equip the automorphism group of a rooted tree with a natural metric and define a family of contractions on $Aut(X^{\ast})$. Moreover, we construct an iterated function system (IFS) whose attractor is the closure of the adding machine group on $Aut(X^{\ast})$. Then we particulary investigate self-similarity of compact topological groups in the sense of IFS. We also look into relations between profinite groups and self-similar groups in the sense of IFS. Finally, we show that some subgroups of the group of $p-$adic numbers $\mathbb{Q}_{p}$ are self-similar in the sense of IFS and build some examples.