Proportion of periodic points in reduction of polynomials

Technical Summary: Proportion of Periodic Points in Reduction of Polynomials

1. Problem Statement
The paper addresses a fundamental question in arithmetic dynamics posed by Juul, Kurlberg, Madhu, and Tucker (2014). Given a number field $K$ and a rational function $f$ with coefficients in $K$, let $f_{\mathfrak{p}}$ denote the reduction of $f$ modulo a prime ideal $\mathfrak{p}$ of the ring of integers $\mathcal{O}_K$. Since the residue field $\mathbb{F}_{\mathfrak{p}} = \mathcal{O}_K/\mathfrak{p}$ is finite, every point is either periodic or strictly preperiodic under iteration.

The central question is to determine the value of:
$$\text{Per}_{\inf}(f, K) := \liminf_{N(\mathfrak{p}) \to \infty} \frac{\# \text{Per}(f, \mathbb{F}_{\mathfrak{p}})}{N(\mathfrak{p})}$$
where $N(\mathfrak{p})$ is the norm of the prime ideal. While random maps suggest this proportion should tend to zero, the specific algebraic structure of polynomials often defies this heuristic. Previous work had classified the case for quadratic polynomials and polynomials with disjoint critical orbits, but a complete classification for all polynomials remained open.

2. Methodology
The author employs a combination of arithmetic dynamics, Galois theory, and algebraic number theory. The methodology proceeds in three main stages:

• Iterated Galois Groups and Fixed-Point Proportion (FPP):
  The paper utilizes the upper bound established by Juul et al.:
  $$\text{Per}_{\inf}(f, K) \leq \text{FPP}(G_{\infty}(\mathbb{C}, f, t))$$
  where $G_{\infty}$ is the iterated Galois group of $f$ over $\mathbb{C}$ and $\text{FPP}$ is the proportion of elements in the profinite group that fix at least one point in every level of the preimage tree.

  • A recent breakthrough by Fariña-Asategui and the author (2024) classifies the FPP for polynomials. It is non-zero if and only if $f$ is linearly conjugate over $\mathbb{C}$ to a Chebyshev polynomial $\pm T_d$. In all other cases, the FPP is zero, implying $\text{Per}_{\inf}(f, K) = 0$.

• Reduction to Chebyshev Polynomials:
  For polynomials $f$ that are linearly conjugate to $\pm T_d$ over $\mathbb{C}$, the upper bound is insufficient. The author proves that $\text{Per}_{\inf}(f, K) = \text{Per}_{\inf}(T_d, K)$, even if the conjugation map $L$ does not have coefficients in $K$.

  • If the conjugation is defined over a quadratic extension $K'/K$, the author analyzes the behavior of prime ideals that split versus those that remain inert in $K'$, showing the limit inferior remains invariant.

• Explicit Calculation via Chebotarev's Density Theorem:
  To calculate $\text{Per}_{\inf}(T_d, K)$, the author derives a formula based on the number of periodic points of $T_d$ over finite fields. This count depends on the divisors of $N(\mathfrak{p}) \pm 1$ that are coprime to $d$.

  • The core technical tool is Proposition 2.5, a consequence of the Chebotarev Density Theorem. It allows the construction of infinitely many prime ideals $\mathfrak{p}$ such that $N(\mathfrak{p})$ satisfies specific congruence conditions modulo $m_K$ (the maximal order of roots of unity in $K$) and other integers.
  • By manipulating these congruences, the author forces the "coprime divisors" $r(N(\mathfrak{p}) \pm 1, d)$ to be small relative to $N(\mathfrak{p})$, thereby determining the limit inferior.

3. Key Contributions and Results
The paper provides a complete classification of $\text{Per}_{\inf}(f, K)$ for any polynomial $f \in K[x]$.

Theorem 1 (Main Classification):
Let $d = \deg(f)$ and $m_K$ be the maximal integer such that $K$ contains a primitive $m_K$-th root of unity.

1. Trivial Cases:
   • If $d=0$, $\text{Per}_{\inf} = 0$.
   • If $d=1$, $\text{Per}_{\inf} = 1$.
2. Non-Chebyshev Cases:
   • If $d \geq 2$ and $f$ is not linearly conjugate over $\mathbb{C}$ to $\pm T_d$, then $\text{Per}_{\inf}(f, K) = 0$.
3. Chebyshev Cases ($f \sim \pm T_d$):
   The value depends on the prime factorization of $d$ and its relationship with $m_K$:
   • Zero Cases:
     • If $d$ has $\geq 2$ prime factors and $\text{rad}(d) \nmid \text{rad}(m_K)$.
     • If $d$ is even, has $\geq 2$ prime factors, $\text{rad}(d) \mid \text{rad}(m_K)$, but $v_2(m_K) = 1$.
   • Non-Zero Cases:
     • If $d$ is an odd prime power: $\text{Per}_{\inf} = 1/2$.
     • If $d$ is an even prime power: $\text{Per}_{\inf} = 1/4$.
     • If $d$ is odd, has $\geq 2$ prime factors, and $\text{rad}(d) \mid \text{rad}(m_K)$: $\text{Per}_{\inf} = 1/2$.
     • If $d$ is even, has $\geq 2$ prime factors, $\text{rad}(d) \mid \text{rad}(m_K)$, and $v_2(m_K) \geq 2$: $\text{Per}_{\inf} = 1/4$.

Corollary 2 (Case $K=\mathbb{Q}$):
Since $m_{\mathbb{Q}} = 2$, the classification simplifies significantly. For $f \in \mathbb{Q}[x]$ with $d \geq 2$:

• $\text{Per}_{\inf} = 1/2$ if $d$ is an odd prime power and $f \sim \pm T_d$.
• $\text{Per}_{\inf} = 1/4$ if $d$ is an even prime power and $f \sim \pm T_d$.
• $\text{Per}_{\inf} = 0$ otherwise.

4. Significance

• Completeness: This work resolves the classification problem for polynomials initiated in 2014, moving beyond specific examples (like quadratics) to a general theory for all degrees and number fields.
• Interplay of Fields: It demonstrates a deep connection between the arithmetic of the base field (specifically the presence of roots of unity, $m_K$) and the dynamical behavior of the reduction of polynomials. The presence of roots of unity in $K$ can "rescue" the proportion of periodic points from zero, even for composite degrees.
• Methodological Advancement: The paper refines the use of iterated Galois groups and fixed-point proportions, showing how to handle cases where the geometric bound is positive by performing explicit arithmetic analysis of the reduction modulo primes.
• Future Directions: The author notes that while the polynomial case is settled, the case for general rational functions remains open, particularly regarding Lattès maps, where the fixed-point proportion of the iterated Galois group is not yet fully understood.