Periodic points in towers of finite fields for polynomials associated to algebraic groups.(English)Zbl 1480.37093

Let $$K$$ be a field, $$\phi(z)$$ a polynomial in $$K(z]$$, $$\phi^n(z)$$ the $$n$$-th iterate of $$\phi$$ under composition and $$\phi^0(z) = z$$. $$\mathcal{O}_\phi(\alpha)$$ denotes the (forward) orbit of a point $$\alpha$$ under $$\phi$$, i.e., $$\{ \phi^n(z) \mid n \geq 0 \}$$, and $$\mathrm{Per}(\phi, K)$$ is the set of periodic points for $$\phi$$ in the field $$K$$, i.e., $$\{ \alpha \in K \mid \phi^n(\alpha) = \alpha \,\,\text{ for some\,\,} n >0 \}$$. Throughout the paper, $$p$$ and $$q$$ represent distinct primes, $$n$$ is a positive integer, and $$v_q(n)$$ is the $$q$$-adic valuation; i.e., if $$n = q^\nu d$$ with $$q \nmid d$$, then $$v_q(n) = \nu$$. $$\delta$$ is the multiplicative order of $$p$$ modulo $$q$$, i.e., the smallest positive integer such that $$q \mid \left(p^\delta - 1\right)$$.

When iterating a polynomial function $$\phi$$ over a finite field, the orbit of any point $$\alpha \in \mathbb{F}_{p^n}$$ is a finite set, that is, all points are preperiodic, meaning the orbit eventually enters a cycle. However, many natural questions about the structure of orbits over finite fields remain:
1
Fix a finite field $$\mathbb{F}_{p^n}$$ and look over all polynomials of fixed degree $$d$$: on average, are there “many” periodic points with relatively small tails leading into the cycles? Or, do we expect few periodic points with long tails?
2
Fix a polynomial defined over $$\mathbb{Q}$$. What is the proportion of periodic points for the reduced map over $$\mathbb{F}_p$$ as $$p \to \infty$$?
3
Again, fix a polynomial: How does the proportion of periodic points in $$\mathbb{F}_{p^n}$$ vary as $$n \to \infty$$?

The work by R. Flynn and D. Garton [Int. J. Number Theory 10, No. 3, 779–792 (2014; Zbl 1309.37083)] addresses the first question. Using combinatorial arguments, they provide upper and lower bounds for the average number of periodic points over all polynomials of degree $$d$$. For $$d$$ large, that is, $$d \geq \sqrt{p^n}$$, their bound of $$(5/6)\sqrt{p^n}$$ agrees with earlier heuristic arguments.
In her thesis [Galois theory and polynomial orbits. Rochester, NY: University of Rochester (Ph.D. dissertation) (2011)], K. Madhu tackles the second question for polynomials $$\phi(z) = z^m + c$$ over $$\mathbb{F}_p$$, using Galois-theoretic methods. With some restrictions on $$c$$, she shows that, for primes congruent to $$1$$ modulo $$m$$, the proportion of points in $$\mathbb{F}_p$$ that are periodic points for $$\phi$$ goes to $$0$$ as $$p \to \infty$$. Her work was later generalized for rational functions by J. Juul et al. [Int. Math. Res. Not. 2016, No. 13, 3944–3969 (2016; Zbl 1404.37124)]. There, they show that, for many rational functions, the proportion of periodic points should be small. M. Sha and S. Hu [Acta Arith. 148, No. 4, 309–331 (2011; Zbl 1272.37040)] fix an $$n$$ and look at power maps over $$\mathbb{F}_{p^n}$$. Exploiting the underlying group structure of such functions allows them to find the number of periodic points for such functions over $$\mathbb{F}_{p^n}$$ and to compute the asymptotic mean number of period points as $$p \to \infty$$.
In this paper, the authors focus on the third question in the special case that the polynomial map $$\phi(z)$$ can be viewed as an endomorphism of an underlying algebraic group.
In Section 4, the authors fix the polynomial $\phi(z) = z^{q},$ for $$q$$ prime, and study the proportion of periodic points in $$\mathbb{F}_{p^n}$$ as $$n$$ grows. In particular, they consider the following limits.
{Definition}. They define the following proportions for integers $$\nu \geq 0$$. $P_{\nu}(\phi) = \lim_{\substack{ n \to \infty\\ \delta \mid n\\ v_q(n)=\nu }}\frac{\# \mathrm{Per}\left(\phi, \mathbb{F}_{p^n}\right)}{p^n}.$
Here are the main results for pure power maps in Section 4:

{Theorem}. Let $$q=2$$, $$v_2(p-1)= \lambda$$ and $$\max\{v_2(p-1), v_2(p+ 1)\} = \mu$$. Then, for $$\phi(z) = z^2$$ we have $P_0(\phi) = \frac 1{2^\lambda}, \qquad P_{\nu}(\phi)=\frac{1}{2^{\mu+\nu}}, \quad \text{for }\nu \geq 1.$
{Theorem}. Let $$q$$ be an odd prime, $$\delta$$ the multiplicative order of $$p$$ modulo $$q$$, and $$v_q(p^\delta-1)= \mu \geq 1$$. For $$\phi(z) = z^q$$, we have $P_\nu(\phi) = \frac 1{q^{\mu+\nu}}.$
In Section 5, the authors consider $$T_q(z)$$, the Chebyshev polynomial of prime degree $$q$$.
{Definition}. They define the following proportions for integers $$\nu \geq 0$$. $R_{\nu}(T_q) = \lim_{\substack{ n \to \infty\\ \delta \mid 2n\\ v_q(n)=\nu }}\frac{\# \mathrm{Per}\left(T_q, \mathbb{F}_{p^n}\right)}{p^n}.$
Here are the main results for Chebyshev polynomials in Section 5:

{Theorem}. Let $$q=2$$ and $$\mu = \max\{v_2(p-1), v_2(p+1)\}$$. Then: $R_\nu(T_2) = \frac{2^{\mu+\nu-1}+1}{2^{\mu+\nu+1}}.$
{Theorem}. Let $$q$$ be an odd prime. Let $$v_q(p^{\delta}-1) = \mu \geq 1$$. Then: $R_\nu(T_q) = \frac{q^{\mu+\nu}+1}{2 q^{\mu+\nu}}.$
They then define the ratios of interest: $R_{\delta,\nu}(T_t) = \lim_{\substack{ n \to \infty\\ \gcd(\Delta, n) = \delta\\ v(n)=\langle \nu_i\rangle}} \frac{\# \mathrm{Per}\left(T_t, \mathbb{F}_{p^n}\right)}{p^n}.$
{Theorem}. Let $$t=q_1^{f_1}q_2^{f_2}\ldots q_r^{f_r}$$, with $$q_i$$ distinct odd primes for $$1\leq i\leq r.$$ Then, there are disjoint subsets $$I, J \subseteq \{1, 2, \ldots, r \}$$, such that $R_{\delta,\nu}(T_t) = \frac{Q_{I} + Q_{J}}{2Q_IQ_J},$ where $Q_I = \prod_{i\in I}{q_i^{\mu_i + \nu_i}} \quad \text{ and } \quad Q_{J} = \prod_{j\in J}{q_j^{\mu_j+ \nu_j}}.$

MSC:

 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 37P35 Arithmetic properties of periodic points 37P25 Dynamical systems over finite ground fields 11T06 Polynomials over finite fields

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