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:
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?
Fix a polynomial defined over \(\mathbb{Q}\). What is the proportion of periodic points for the reduced map over \( \mathbb{F}_p\) as \(p \to \infty\)?
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 \[ Q_I = \prod_{i\in I}{q_i^{\mu_i + \nu_i}} \quad \text{ and } \quad Q_{J} = \prod_{j\in J}{q_j^{\mu_j+ \nu_j}}. \]


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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