Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ${\mathbb{Z}}_d$ of $d$-adic integers

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.