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Dynamical Mordell-Lang conjecture for split self-maps of affine curve times projective curve

This paper proves the dynamical Mordell-Lang conjecture for split self-maps on the product of an affine curve and a projective curve defined over the algebraic numbers.

Original authors: Junyi Xie, She Yang, Aoyang Zheng

Published 2026-04-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Junyi Xie, She Yang, Aoyang Zheng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a complex dance performance on a stage that has two distinct zones: a flat, open floor (like an infinite parking lot) and a curved, circular track (like a racetrack that loops back on itself).

In this paper, mathematicians Junyi Xie, She Yang, and Aoyang Zheng are studying a specific question about how dancers move on this combined stage.

The Big Question: The "Mordell–Lang" Dance Rule

The core problem they are solving is called the Dynamical Mordell–Lang Conjecture. Here is the plain English version:

Imagine a dancer starts at a specific spot. Every second, they follow a strict set of rules to jump to a new spot (this is called an "iteration"). There is also a specific line drawn on the stage (a "curve").

The conjecture asks: If we watch this dancer forever, how often will they land exactly on that line?

The mathematicians suspect the answer is always simple: The times the dancer lands on the line will form a predictable pattern, like a clock ticking at regular intervals (e.g., every 5th second, or every 10th second), or they will stop landing on the line entirely after a while. They will never land on the line in a chaotic, random, or "messy" way.

The Stage Setup: Affine vs. Projective

The authors are looking at a specific type of stage:

Zone A (The Affine Curve): Think of this as an infinite straight road. It goes on forever in one direction.
Zone B (The Projective Curve): Think of this as a circle or a loop. It's finite but has no "ends" because it connects back to itself.

The dancers move on the combination of these two: (Infinite Road) × (Circular Track).

The dancer moves along the road according to Rule A.
The dancer moves along the track according to Rule B.
The two movements happen simultaneously.

The Main Discovery

The authors proved that for this specific combination of an infinite road and a circular track, the "Mordell–Lang Dance Rule" is always true.

No matter how complicated the rules are for the dancer's jumps, if you draw a line on this combined stage, the dancer will either:

Hit the line at regular, predictable intervals forever.
Hit the line a few times and then never again.

They will never hit the line in a weird, unpredictable pattern.

How Did They Prove It? (The "Speed" Analogy)

The proof relies on a clever observation about speed.

Imagine the dancer on the Infinite Road is running away from home. Because the road is infinite and the rules are set up a certain way, this runner eventually starts sprinting faster and faster, heading toward the horizon. Their speed increases exponentially (1, 10, 100, 1000...).

Now, look at the dancer on the Circular Track. Even though they are moving, they are stuck on a loop. They can't run away to infinity. No matter how many times they go around, they stay within a fixed distance.

The Conflict:
The authors realized that if the dancer is supposed to land on a specific diagonal line (a line that cuts across both the road and the track), the two parts of their movement have to "sync up" perfectly.

The road-part is zooming off to infinity at a breakneck, exponential speed.
The track-part is just spinning around.

The authors showed that it is mathematically impossible for the "track-part" to keep up with the "road-part" in a way that allows them to hit a specific diagonal line over and over again, unless the line is perfectly aligned with the road or the track itself.

If the line is tilted (diagonal), the road-runner will zoom past the line so fast that the track-runner can never catch up to meet it again. The "intersection" (the moment they land on the line) becomes a finite event.

Why Does This Matter?

This is a big deal in the world of Arithmetic Dynamics (a mix of number theory and chaos theory).

Predictability: It tells us that even in systems that look chaotic, there are hidden structures that force patterns to be simple and regular.
Building Blocks: The authors showed that if you understand the rules for simple shapes (like a line and a circle), you can understand much more complex shapes built from them.
The "Split" Concept: They focused on "split" maps, meaning the two zones move independently. This is like proving that if you know how a car drives on a highway and how a boat sails on a lake, you can predict what happens if you put the car on a boat.

Summary

In short, these mathematicians proved that on a stage made of an infinite road and a circular track, a dancer following strict rules cannot land on a diagonal line in a messy, random way. They will either land on it like a metronome (predictable) or stop landing on it entirely. The "infinite speed" of the road eventually outpaces any attempt to stay synchronized with the circular track, forcing the pattern to be simple.

1. Problem Statement

The paper addresses the Dynamical Mordell–Lang (DML) Conjecture, a central problem in arithmetic dynamics proposed by Ghioca and Tucker.

The Conjecture: Let $f: X \to X$ be an endomorphism of a quasi-projective variety over a field $K$ of characteristic 0, and let $V \subset X$ be a closed subvariety. For any point $x \in X(K)$ , the return set $\{n \in \mathbb{N} \mid f^n(x) \in V(K)\}$ is a finite union of arithmetic progressions.
Specific Focus: The authors investigate this conjecture for split self-maps (product maps) of the form $f \times g$ $f \times g$ acting on the product space $X \times Y$ $X \times Y$ , where:
- $X$ is an affine curve.
- $Y$ is a projective curve.
- The base field is $\mathbb{Q}$ .

While the conjecture is known for étale maps and for the affine plane $\mathbb{A}^2$ , the case of mixed products (affine $\times$ projective) presents unique challenges regarding the interaction of growth rates in different geometric components.

2. Methodology

The proof strategy relies on a combination of arithmetic geometry, local height analysis, and dynamical reduction techniques.

A. Reduction to the Core Case

The authors reduce the general problem (Theorem 1.1) to the specific case of Proposition 1.2:

$X = \mathbb{A}^1$ (affine line) and $Y = \mathbb{P}^1$ (projective line).
They assume $\deg(f) = \deg(g) = d > 1$ .
They assume no iterate of $g$ is conjugate to a polynomial map (to avoid trivial reductions to the $\mathbb{A}^2$ case).
They assume the starting point $(x_0, y_0)$ consists of non-preperiodic points and the curve $C$ is not a fiber.

B. Local Height and Growth Rate Analysis

The core technical innovation lies in comparing the asymptotic behavior of orbits in the affine coordinate versus the projective coordinate using local heights (valuations).

Affine Growth (Lemma 2.2): For a non-preperiodic point $x_0$ under $f: \mathbb{A}^1 \to \mathbb{A}^1$ , there exists a place $v$ where the orbit $|f^n(x_0)|_v$ grows at the maximal exponential rate $d^n$ . Specifically, $c_1 d^n < \log |f^n(x_0)|_v < c_2 d^n$ .
Projective Stability (Lemma 2.1): Citing results from Silverman ([Sil93]), for a non-exceptional point $p \in \mathbb{P}^1$ , a non-preperiodic orbit cannot approach $p$ with the maximal speed $d^n$ . The limit of the normalized local height is 0.

C. The Contradiction Argument

The proof proceeds by contradiction:

Assume the intersection of the orbit of $(x_0, y_0)$ with a curve $C$ is infinite.
By Lemma 2.2, the affine coordinate $x_n$ tends to infinity at speed $d^n$ in some valuation $v$ .
The defining equation of the curve $C$ relates $x_n$ and $y_n$ . As $x_n \to \infty$ , the equation forces the projective coordinate $y_n$ to approach specific points (roots of the leading coefficient of the curve's equation at infinity) extremely rapidly.
Specifically, the distance $|y_n - b|_v$ must decay faster than $d^{-n}$ (i.e., $-\log |y_n - b|_v > c d^n$ ).
However, Lemma 2.1 states that for a non-exceptional point $b$ , the orbit $y_n$ cannot approach $b$ with such maximal speed.
The authors verify that under their assumptions (no iterate of $g$ is a polynomial), $g$ has no exceptional points, rendering the rapid convergence impossible. Thus, the intersection must be finite.

3. Key Contributions

Proof of Theorem 1.1: The DML conjecture is proven for $X \times Y$ where $X$ is an affine curve and $Y$ is a projective curve over $\mathbb{Q}$ .
Proof of Proposition 1.2: Establishes the result for the fundamental case of $\mathbb{A}^1 \times \mathbb{P}^1$ , which serves as the building block for the general theorem.
Generalization to DML(1): The authors prove Corollary 1.3, showing that if $(X, f)$ and $(Y, g)$ satisfy the DML(1) property (the conjecture holds for curves), then their product $(X \times Y, f \times g)$ also satisfies DML(1). This extends the result to higher-dimensional varieties where the subvariety is a curve.
Handling Mixed Geometries: The paper successfully bridges the gap between affine dynamics (where points escape to infinity) and projective dynamics (where points may be bounded or approach specific points), resolving the conflict in growth rates that arises in product spaces.

4. Results

Main Theorem: For an affine curve $X$ and a projective curve $Y$ over $\mathbb{Q}$ , and endomorphisms $f, g$ , the return set of any point under $f \times g$ to any closed subvariety is a finite union of arithmetic progressions.
Classification by Genus: The proof of Theorem 1.1 utilizes a case-by-case analysis based on the genus of the projective closures of $X$ $X$ and $Y$ $Y$ :
- Genus $> 1$ : Trivial (finite orbits).
- Genus $= 1$ : Étale maps (known results apply).
- Genus $= 0$ : Reduces to the $\mathbb{A}^1 \times \mathbb{P}^1$ case handled in Proposition 1.2.

5. Significance

Advancement of Arithmetic Dynamics: This work resolves a significant open case of the DML conjecture, moving beyond the previously known cases of étale maps and pure affine spaces.
Methodological Impact: The technique of comparing local height growth rates between different factors of a product space provides a powerful new tool for analyzing split dynamical systems. It demonstrates how "fast" growth in one coordinate can force "impossible" convergence in another, leading to finiteness results.
Foundation for Higher Dimensions: By establishing the DML(1) property for products of curves, the paper lays the groundwork for tackling the conjecture in higher-dimensional varieties constructed from curves, a crucial step toward the general conjecture.

In summary, Xie, Yang, and Zheng provide a rigorous proof that the dynamical Mordell–Lang conjecture holds for products of affine and projective curves, utilizing a sophisticated analysis of local heights to show that orbits cannot intersect arbitrary curves infinitely often unless they follow specific arithmetic progressions.