The Asymptotic Number of Periodic Points of Discrete $p$-Adic Dynamical Systems

Let $A(n,a,y)$ denote a specific weighted average of different zeros of $f^n(x)-x$ for all prime numbers $p\leq y$, where $f(x)=x^p+ax\in\mathbb{F}_p[x]$, $a\neq 0$, and $f^n$ denotes the $n$-fold composition of $f$ by itself. If $a=1$, then $A(n, a, x)\to 0$ as $x\to\infty$, and if $a>1$, then $A(n,a,x) \to 1$ as $x \to \infty$. We also discuss a method for counting the number of linear factors of a polynomial whose zeros are $n$-periodic points of $f(x)\in\mathbb Z[x]$ by using a theorem of Frobenius. Finally, we obtain some results in the monomial case over $p$-adic numbers by using this method.