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

Elsayed Ahmed; Dmytro Savchuk

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arXiv · 1711.06735

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

Elsayed Ahmed·Dmytro Savchuk

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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.

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BibTeX RIS

Elsayed Ahmed, Dmytro Savchuk. 2017-11-17. Endomorphisms of regular rooted trees induced by the action of polynomials on the ring $\mathbb Z_d$ of $d$-adic integers. https://arxiv.org/abs/1711.06735

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