Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ℤd of d-adic integers

Abstract

We show that every polynomial in [Formula: see text] defines an endomorphism of the [Formula: see text]-ary rooted tree induced by its action on the ring [Formula: see text] of [Formula: see text]-adic integers. The sections of this endomorphism also turn out to be induced by polynomials in [Formula: see text] of the same degree. In the case of permutational polynomials acting on [Formula: see text] by bijections, the induced endomorphisms are automorphisms of the tree. In the case of [Formula: see text], such polynomials were completely characterized by Rivest in [Permutation polynomials modulo [Formula: see text], Finite Fields Appl. 7(2) (2001) 287–292]. As our main application, we utilize the result of Rivest to derive the condition on the coefficients of a permutational polynomial [Formula: see text] that is necessary and sufficient for [Formula: see text] to induce a level transitive automorphism of the binary tree, which is equivalent to the ergodicity of the action of [Formula: see text] on [Formula: see text] with respect to the normalized Haar measure.