[Paper Review] On Degree Growth and Stabilization of Three Dimensional Monomial Maps

Jan-Li Lin|arXiv (Cornell University)|Apr 27, 2012

Mathematical Dynamics and Fractals4 references4 citations

TL;DR

This paper develops tools to analyze degree growth and stabilization of three-dimensional monomial maps, classifying their dynamical behavior via cohomological stability. It proves that for diagonalizable 3D monomial maps with eigenvalues whose ratios are roots of unity, a complete simplicial refinement of the fan ensures eventual 1- and 2-stability under iteration, resolving long-standing questions in higher-dimensional dynamics.

ABSTRACT

In this paper, we develop several tools to study the degree growth and stabilization of monomial maps. Using these tools, we can classify semisimple three dimensional monomial maps by their dynamical behavior.

Motivation & Objective

• To understand the degree growth and stabilization behavior of three-dimensional monomial maps.
• To classify semisimple 3D monomial maps by their dynamical behavior using cohomological stability.
• To determine when monomial maps admit stable models via toric modifications.
• To resolve cases where 1-stability and 2-stability may not coexist or be achievable via blowups.
• To analyze the failure of stabilization in cases with complex eigenvalues whose ratio is not a root of unity.

Proposed method

• Uses the pullback action on cohomology groups $H^{p,p}(X)$ to define $p$-stability for rational maps on toric varieties.
• Applies toric geometry tools: fans, rational polyhedral cones, and refinements to construct birational models.
• Analyzes the action of the monodromy matrix $A$ on rays and cones to track asymptotic behavior under iteration.
• Applies Lemma 6.1 to detect instability when rays converge toward a cone’s interior or face.
• Employs case analysis based on the location of eigenspaces relative to the fan structure and cone types.
• Uses rational conjugation to reduce dynamics to rotation-like behavior with irrational angles, proving instability in Case 3.

Experimental results

Research questions

• RQ1 Under what conditions on the eigenvalues of a 3D diagonalizable monomial map does a 1- and 2-stable model exist after toric modification?
• RQ2 Can a monomial map be 1-stable but not 2-stable, or vice versa, on any smooth projective toric variety?
• RQ3 Is it possible for a monomial map to have no 1-stable or 2-stable model, even after birational modification?
• RQ4 What role does the ratio $\mu/\bar{\mu}$ of complex eigenvalues play in determining the existence of stable models?
• RQ5 Can the stabilization process fail even when the map is stable on the original toric variety?

Key findings

• For 3D diagonalizable monomial maps where $\mu/\bar{\mu}$ is a root of unity for all eigenvalues $\mu$, there exists a complete simplicial refinement $\Delta'$ such that $f_A^k$ is both 1-stable and 2-stable for all $k \geq k_0$.
• If eigenvalues have distinct moduli, the stable model can be chosen to be smooth and projective.
• When $\{A^k\}$ is finite, $f_A$ becomes an automorphism on a smooth projective toric variety.
• In the case where two complex eigenvalues $\mu, \bar{\mu}$ have $\mu/\bar{\mu}$ not a root of unity, $f_A$ may lack both 1- and 2-stable models.
• For such cases, $f_A$ can be 1-stable on a smooth variety but not 2-stable, or vice versa, and may not become stable via blowups.
• In Case 3 where $|\mu| = |\nu|$, the map cannot be made 1- or 2-stable on any complete toric variety due to irrational rotation dynamics.