Endomorphisms of regular rooted trees induced by the action of polynomials on the ring $\mathbb Z_d$ of $d$-adic integers

TL;DR

This paper explores how polynomials over the ring of $d$-adic integers induce endomorphisms on regular rooted trees, characterizing when these are automorphisms and their ergodic properties, especially for the binary case.

Contribution

It establishes that polynomials in $ extbf{Z}[x]$ induce endomorphisms of $d$-ary rooted trees and characterizes when these are automorphisms and level transitive, extending Rivest's results.

Findings

Polynomials induce endomorphisms of $d$-ary rooted trees.

Characterization of automorphisms induced by permutational polynomials.

Necessary and sufficient conditions for ergodicity of the induced automorphisms.

Abstract

We show that every polynomial in $\mathbb Z[x]$ defines an endomorphism of the $d$-ary rooted tree induced by its action on the ring $\mathbb Z_d$ of $d$-adic integers. The sections of this endomorphism also turn out to be induced by polynomials in $\mathbb Z[x]$ of the same degree. In the case of permutational polynomials acting on $\mathbb Z_d$ by bijections the induced endomorphisms are automorphisms of the tree. In the case of $\mathbb Z_2$ such polynomials were completely characterized by Rivest. As our main application we utilize the result of Rivest to derive the condition on the coefficients of a permutational polynomial $f(x)\in\mathbb Z[x]$ that is necessary and sufficient for $f$ to induce a level transitive automorphism of the binary tree, which is equivalent to the ergodicity of the action of $f(x)$ on $\mathbb Z_2$ with respect to the normalized Haar measure.