On conjugacy in groups of finite-state automorphisms of rooted trees

We show that the conjugacy of elements of finite order in the group of finite-state automorphisms of a rooted tree is equivalent to their conjugacy in the group of all automorphisms of the rooted tree. We establish a criterion for conjugacy between a finite-state automorphism and the adding machine in the group of finite-state automorphisms of a rooted tree of valency 2. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1357–1366, October, 2008.