[Paper Review] Iteration and the Minimal Resultant

Kenneth Jacobs, Phillip Williams|arXiv (Cornell University)|Aug 6, 2016

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper characterizes rational maps over non-Archimedean fields for which the minimal resultant of iterates follows a simple power-law formula in terms of degree and iterate count. Using geometric invariant theory and Berkovich space dynamics, it shows this occurs precisely when the map has semi-stable reduction for all iterates and reduction outside the indeterminacy locus $I(d)$, with applications to Arakelov-Green's functions and Lattès maps.

ABSTRACT

Let $K$ be an algebraically closed field that is complete with respect to a non-Archimedean absolute value, and let $φ\in K(z)$ have degree $d\geq 2$. We characterize maps for which the minimal resultant of an iterate $φ^n$ is given by a simple formula in terms of $d$, $n$, and the minimal resultant of $φ$. We show that such maps are precisely those with reduction outside of an indeterminacy locus $I(d)$ and which also have semi-stable reduction for every iterate $φ^n$. We give two equivalent ways of describing such maps, one measure theoretic and the other in terms of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps. As an application, we are able to give an explicit formula for the minimal value of the diagonal Arakelov-Green's function of a map satisfying the conditions of the main theorem. We illustrate our results with some explicit calculations in the case of the Lattès maps.

Motivation & Objective

• To determine when the minimal resultant of an iterate $\varphi^n$ satisfies a simple formula in terms of $d$, $n$, and $R_{[\varphi]}$.
• To identify the dynamical and geometric conditions under which the minimal resultant behaves predictably under iteration.
• To connect the minimal resultant locus with the equilibrium measure and barycenter on the Berkovich projective line.
• To apply the results to compute the minimal value of the diagonal Arakelov-Green's function for such maps.
• To provide explicit examples, including Lattès maps, illustrating both cases where the formula holds and where it fails.

Proposed method

• Uses normalized lifts of rational maps and defines the resultant valuation $R_\varphi$ as a conjugacy invariant.
• Applies geometric invariant theory to relate minimality of the resultant to semi-stable reduction and the indeterminacy locus $I(d)$.
• Employs Berkovich space techniques to analyze the equilibrium measure $\mu_\varphi$ and its barycenter.
• Introduces two 'residue' measures from DeMarco and others to study reduction behavior under iteration.
• Establishes equivalence between the minimal resultant formula and two geometric conditions: stability of the minimal resultant locus and absence of barycenter points in $I(d)$.
• Applies results to compute the minimal diagonal Arakelov-Green's function value via explicit calculations on Tate curves and Lattès maps.

Experimental results

Research questions

• RQ1 Under what conditions does the minimal resultant of $\varphi^n$ satisfy $R_{[\varphi^n]} = \frac{d^n(d^n-1)}{d(d-1)} R_{[\varphi]}$?
• RQ2 How do semi-stable reduction and the indeterminacy locus $I(d)$ influence the behavior of the minimal resultant under iteration?
• RQ3 What is the relationship between the barycenter of the equilibrium measure $\mu_\varphi$ and the minimal resultant locus?
• RQ4 Can the minimal value of the diagonal Arakelov-Green's function be explicitly computed for maps satisfying the minimal resultant formula?
• RQ5 Why do Lattès maps with even-degree multiplication fail to satisfy the minimal resultant formula, while odd-degree ones do?

Key findings

• The minimal resultant of $\varphi^n$ satisfies the simple formula $R_{[\varphi^n]} = \frac{d^n(d^n-1)}{d(d-1)} R_{[\varphi]}$ if and only if $\varphi$ has semi-stable reduction for all iterates and its reduction lies outside the indeterminacy locus $I(d)$.
• For Lattès maps $\psi_m$ with odd $m$, the minimal resultant formula holds; for even $m$, it fails due to changing minimal resultant loci across iterates.
• The minimal value of the diagonal Arakelov-Green's function for $\varphi_2$ (multiplication-by-2 on a Tate curve) is $\frac{1}{12}\log_v|q|$, computed via Néron-Tate height and Bernoulli polynomials.
• The minimal resultant locus of $\varphi$ is contained in that of $\varphi^n$ for all $n$ when the equivalent conditions hold, as in the case of $\varphi(z) = \frac{z^p - z}{p}$ over $\mathbb{C}_v$.
• The equilibrium measure $\mu_\varphi$ for $\varphi(z) = \frac{z^p - z}{p}$ is Haar measure on $\mathbb{Z}_p$, and its barycenter corresponds to a conjugate with reduction outside $I(d)$, satisfying the theorem's conditions.
• Explicit calculations confirm that the minimal resultant formula fails for even-degree Lattès maps because the minimal resultant locus shifts with each iterate, preventing a uniform conjugate with semi-stable reduction.