The asymptotic number of periodic points of discrete polynomial p-adic dynamical systems

Marcus Nilsson         Robert Nyqvist

ABSTRACT

Let $\Phi_{n, f}(x)$ be the polynomial whose zeros are the nth periodic points of the polynomial $f(x) \in \mathbb{Z} [x]$. We will show that if $\Phi_{n, f}(x)$ is separable over $\mathbb{Q}$, then the average of numbers of linear factors of $\Phi_{n, f}(x)$ modulo $p\mathbb{Z} [x]$ is equal to the number of orbits when the Galois group of $\Phi_{n, f}$ acts on the set of zeros of $\Phi_{n, f}(x)$. The Cebotarev Density Theorem and its application to determine the Galois group of a polynomial is discussed. The wreath product is introduced as an possible representation of the Galois group of $\Phi_{n, f}$. If f(x) = xⁿ, where $n \geqslant 2$, then we are able to lift the result to the p-adic numbers, and explicit expressions are found of the average of numbers of linear factors of $\Phi_{n, f}(x)$ over the p-adic integers.

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