Height arguments toward the dynamical Mordell-Lang problem in arbitrary characteristic

Technical Summary: Height Arguments toward the Dynamical Mordell–Lang Problem in Arbitrary Characteristic

Problem Statement
The paper addresses the Dynamical Mordell–Lang (DML) conjecture, a central problem in arithmetic dynamics. The conjecture posits that for a rational self-map $f$ of a variety $X$ and a point $x$ with a well-defined orbit, the return set $\{n \in \mathbb{N} \mid f^n(x) \in V\}$ (where $V$ is a closed subvariety) is a finite union of arithmetic progressions. While this "0-DML" conjecture holds in characteristic 0 for specific cases (e.g., étale endomorphisms, endomorphisms of $\mathbb{A}^2$), it fails in positive characteristic $p > 0$. In positive characteristic, return sets can exhibit complex structures, leading to the "pDML" problem, which seeks to characterize the form of these sets. Previous work by Corvaja, Ghioca, Scanlon, and Zannier [6] and the authors [37] established that return sets for certain systems are "p-normal" or "widely p-normal" sets. However, the general structure of return sets for arbitrary endomorphisms in positive characteristic remains largely unknown and potentially more complex than previously conjectured.

Methodology
The authors employ height arguments to investigate the DML problem in arbitrary characteristic. The core strategy involves identifying two distinct "speeds" of height growth for the orbit of a point under different line bundles. If the orbit intersects a subvariety infinitely often, the height growth rates calculated via these different bundles must be compatible. The authors demonstrate that under specific dynamical conditions, these growth rates are incompatible, forcing the intersection to be finite.

Key technical tools include:

1. Cohomological Lyapunov Multipliers ($\mu_i(f)$): Introduced by the first author [29, 35], these are defined as ratios of dynamical degrees ($\lambda_i(f)/\lambda_{i-1}(f)$). They serve as eigenvalues of the pullback map on the numerical class group. The authors utilize the condition that these multipliers are not roots of unity (specifically, not in the set $\text{Root} = \{a^{1/n} \mid a, n \in \mathbb{Z}^+\}$) to generate distinct growth rates.
2. Galois Conjugates: To construct the necessary second line bundle for the height argument, the authors utilize Galois conjugates of the cohomological Lyapunov multipliers. The assumption that $\mu_i(f) \notin \text{Root}$ ensures that a conjugate $\mu'$ exists with $|\mu| \neq |\mu'|$, creating the required disparity in growth speeds.
3. Albanese Maps: For general projective varieties, the authors reduce the problem to extremal cases (where the Picard number is trivial or the variety is an abelian variety) using Albanese maps.
4. Moriwaki Heights: While the primary proofs are conducted in positive characteristic using Weil heights, the authors note that the arguments extend to characteristic 0 by substituting the Weil height machinery with Moriwaki's height machinery [22], which satisfies the Northcott property for arithmetic function fields.

Key Contributions and Results

1. Finiteness of Return Sets for Dense Orbits (Theorem 1.1):
   The authors prove that for a surjective endomorphism $f$ of a projective variety $X$, if the cohomological Lyapunov multipliers $\mu_i(f)$ are not in $\text{Root}$ for all $i$, then the return set of a dense orbit into any closed subcurve $C$ is finite. This result generalizes previous findings on cohomologically hyperbolic automorphisms of surfaces [34] to arbitrary dimensions and varieties.

   • Corollary 1.3: This extends to cases where the set of multipliers has an empty intersection with the interval $[1, \deg(f)]$, including cohomologically hyperbolic automorphisms and amplified automorphisms.

2. Split Endomorphisms and Product Systems (Theorem 1.8):
   The paper establishes a result for split endomorphisms of product spaces $X \times C$. If $f: X \to X$ and $g: C \to C$ are surjective endomorphisms such that the degree of $g$ exceeds the first dynamical degree of $f$ ($\deg(g) > \lambda_1(f)$), and if $(X, f)$ satisfies the DML property of type $\epsilon$ (where $\epsilon \in \{0, 1, 2\}$), then the product system $(X \times C, f \times g)$ also satisfies the DML property of type $\epsilon$.

   • Mechanism: The condition $\deg(g) > \lambda_1(f)$ creates a "fast" factor ($C$) and a "slow" factor ($X$). The height argument forces the orbit component in the fast factor to be preperiodic, reducing the problem to the slower factor.
   • Corollary 1.9: Applied to products of curves with distinct degrees, this implies that if the degrees of the factors are strictly increasing (with the first being 1), the system satisfies the DML2 property (widely p-normal sets).

3. Counterexamples and Complexity of Return Sets (Section 5):
   The authors demonstrate that the height argument cannot be applied to all systems, specifically those with zero entropy or int-amplified endomorphisms (which are "isotropic" in terms of growth).

   • Proposition 1.12: They construct an automorphism of a linear torus with zero entropy that does not satisfy the DML1 property. This disproves a conjecture by Ghioca [9] regarding affine varieties.
   • Examples 5.5 and 5.6: The authors provide heuristic and rigorous examples showing that return sets can be significantly more complex than "widely p-normal sets." Specifically, they construct systems where the return set involves solutions to systems of polynomial-exponential equations, yielding sets of the form $\{p^{m_1} + p^{m_2} + p^{m_1+m_2}\}$, which lie outside the scope of previously defined p-normal sets.

Significance and Claims
The paper claims to advance the understanding of the DML problem in positive characteristic by:

• Providing a unified height-based approach that works in arbitrary characteristic (with appropriate height machinery).
• Identifying a broad class of dynamical systems (those with non-root Lyapunov multipliers) where the return sets are finite, extending known results from surfaces to higher dimensions.
• Clarifying the structural relationship between split dynamical systems and their factors, showing how degree disparities control the complexity of return sets.
• Challenging the prevailing intuition that return sets in positive characteristic are bounded by "widely p-normal" structures. The authors explicitly state that their examples show return sets can be "very complicated — more complicated than experts once imagined," even for zero-entropy endomorphisms of tori.

The authors remain modest regarding the full scope of the pDML problem, noting that the case of endomorphisms of tori is equivalent to solving hard Diophantine equations [6] and that their counterexamples suggest the need for a refined conjecture (Question 5.9) involving semilinear sets and systems of equations. They do not claim to have solved the general pDML problem but rather delineate the boundaries where height arguments succeed and where the problem exhibits unprecedented complexity.