Arithmetic dynamics — finite orbits over number fields

Parent: orbit · Prereq: discrete-dynamical-systems

The number-theoretic branch (Silverman's school): iterate a rational map $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ of degree $d \ge 2$ defined over a number field $K$ (think: $\varphi(z) = z^2 + c$ with $c \in \mathbb{Q}$), and ask which rational points have small orbits.

Definition

A point $x$ is preperiodic if its orbit is finite (eventually periodic). Write $\operatorname{Preper}(\varphi, K)$ for the preperiodic points with coordinates in $K$.

The height machine

The tool that makes this number theory rather than dynamics: the Weil height $h(x)$, measuring the arithmetic complexity of a rational point ($h(\tfrac{p}{q}) = \log\max(|p|, |q|)$ in lowest terms). Two facts:

1. $h(\varphi(x)) = d \cdot h(x) + O(1)$ — applying $\varphi$ multiplies complexity by the degree, up to bounded error.
2. Northcott's theorem: only finitely many points of $K$ have height below any given bound.

Canonical height (Call–Silverman)

$\hat{h}_\varphi(x) = \lim_{n \to \infty} \dfrac{h(\varphi^n(x))}{d^{\,n}}$ — the limit exists because fact 1 makes the sequence Cauchy (telescoping the $O(1)$ errors gives $|\hat{h} - h| \le C$ for a constant $C$ independent of $x$). It repairs fact 1 into an exact identity: $\hat{h}_\varphi(\varphi(x)) = d \cdot \hat{h}_\varphi(x)$.

Finite orbits = zero canonical height

$\hat{h}_\varphi(x) = 0 \iff x$ is preperiodic. ($\Leftarrow$: a finite orbit has bounded $h$, so $h(\varphi^n x)/d^n \to 0$. $\Rightarrow$: $\hat{h} = 0$ forces $h(\varphi^n x) \le C$ for all $n$, and Northcott makes the orbit finite.) Corollary: $\operatorname{Preper}(\varphi, K)$ is finite for every $\varphi$ of degree $\ge 2$ over every number field.

Contrast this with dynamics over $\mathbb{R}$ or $\mathbb{C}$, where periodic points are typically dense in the Julia set (complex-dynamics): arithmetic makes finite orbits rare and finite in number. The big open problem is uniformity — Morton–Silverman conjecture: the size of $\operatorname{Preper}(\varphi, K)$ is bounded by a constant depending only on $d$, $\deg K$ (not on $\varphi$). Even for $z^2 + c$ over $\mathbb{Q}$ this is open (Poonen's conjecture: at most 9 rational preperiodic points; periods 4 and 5 are proven impossible, period ≥ 6 partially).

Count along: $\varphi(z) = z^2$ over $\mathbb{Q}$

• $h(2) = \log 2$, orbit $2 \to 4 \to 16 \to 256$: heights $\log 2, 2\log 2, 4\log 2, \dots$ — doubling exactly, so $\hat{h}(2) = \log 2 > 0$: infinite orbit. ✓
• Orbit of $\tfrac{2}{3}$: $\tfrac{2}{3} \to \tfrac{4}{9} \to \tfrac{16}{81}$, heights $\log 3, \log 9, \log 81$ — again doubling; $\hat{h} = \log 3$.
• $\operatorname{Preper}(\varphi, \mathbb{Q}) = \{0, \infty, 1, -1\}$ ($-1 \to 1 \to 1$): four points, exactly the height-zero rationals. Over $\overline{\mathbb{Q}}$ the preperiodic points are $0, \infty$ and all roots of unity — infinitely many, but each individual field cuts out finitely many (Northcott needs bounded degree).

Why this machinery cannot see Collatz

The height argument runs on $h \circ \varphi \approx d \cdot h$ with $d \ge 2$ — a multiplicative complexity growth coming from a globally defined algebraic map. The Collatz map is piecewise affine ($d = 1$ on each branch, and the branches are glued by congruence conditions, not algebra), so canonical heights collapse and Northcott gains no purchase. That structural mismatch — not lack of effort — is why collatz-orbit-statistics runs on measure theory and probability instead.