On confinal dynamics of rooted tree automorphisms (Q2702056)

Let \(T\) be a rooted tree where the valency of every vertex is finite. \(T\) is called spherically homogeneous if any vertex on the \(i\)-th level is connected to \(n_i\) vertices on the \((i+1)\)-st level, where \(n_i\) does not depend on the vertex of the level. Such a tree is uniquely defined up to isomorphism by the sequence \((n_1,n_2,\dots)\).NEWLINENEWLINENEWLINEA ray is any infinite path beginning at the root. The set of rays is called the boundary \(B_T\) of the tree. In a natural way \(B_T\) can be represented by infinite sequences \(x=(x_1,x_2,\dots)\), where \(x_i\in X_i\) and \(|X_i|=n_i\).NEWLINENEWLINENEWLINEIn this paper the authors study the automorphism group of \(T\) by means of their actions on the boundary \(B_T\). Sequences \(x\) and \(y\) are said to be \(s\)-cofinal if for all \(k\geq s\) we have \(x_k=y_k\). Sequences \(x\) and \(y\) are called cofinal when they are \(s\)-cofinal for some \(s\). An automorphism \(\pi\) of \(T\) is called finitary (weakly finitary) if \(x^\pi\) and \(x\) are \(s\)-cofinal (cofinal) for all sequences \(x\) in the boundary. When \(x^\pi\) and \(x\) differ in at most \(s\) places for all \(x\), this \(s\) is called the width of \(\pi\).NEWLINENEWLINENEWLINEThe authors prove that every automorphism of a rooted tree is conjugate to a weakly finitary automorphism of width at most 2, any finite subgroup is conjugate to some subgroup of weakly finitary automorphisms and any automorphism of an \(n\)-regular rooted tree is a product of two automorphisms of finite order not greater than \(n\)!NEWLINENEWLINEFor the entire collection see [Zbl 0940.00028].