Endomorphisms of regular rooted trees induced by the action of polynomials on the ring $\mathbb Z_d$ of $d$-adic integers
| Autor: | Ahmed, Elsayed, Savchuk, Dmytro |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 16 pages, 4 figures |
| Databáze: | arXiv |
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