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A gap principle for polynomial volume growth of zero-entropy automorphisms

This paper establishes a gap principle for the polynomial volume growth of zero-entropy automorphisms on normal projective varieties of dimension $d \ge 4$ , proving that the invariant $\mathrm{plov}(f)$ cannot take values in a specific open interval and thereby determining all possible values in dimension four.

Original authors: Fei Hu, Chen Jiang

Published 2026-04-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Fei Hu, Chen Jiang

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 have a magical machine, a "shape-shifter," that takes a complex geometric object (like a multi-dimensional sculpture) and transforms it over and over again. In mathematics, this is called an automorphism.

Some of these machines are chaotic: they stretch and twist the object so wildly that the complexity explodes exponentially, like a snowball rolling down a hill and becoming a giant boulder in seconds. This is called high entropy.

But this paper is about a very special, calm kind of machine: a zero-entropy automorphism. These machines are gentle. They don't explode the complexity; instead, the complexity grows slowly, like a tree growing a few branches every year. This growth is polynomial (think $n^2$ , $n^3$ , etc., rather than $2^n$ ).

The authors, Fei Hu and Chen Jiang, are asking a very specific question about these gentle machines: "How fast can they grow, and are there any 'forbidden' speeds?"

The Core Concept: The "Growth Meter"

To measure how fast the machine is working, mathematicians use two main rulers:

Degree Growth ( $k$ ): How much the "surface area" of the object stretches with each turn.
Polynomial Volume Growth ($plov$): How much the total "volume" of the path the machine traces out grows.

Think of it like driving a car:

Degree Growth ( $k$ ) is like your speedometer (how fast you are going right now).
Volume Growth ($plov$) is like the total distance you've covered over time, but measured in a specific, complex way related to the shape of the road.

The Big Discovery: The "Gap Principle"

The authors discovered a surprising rule about these machines in dimensions 4 and higher (think of a 4D object, which is hard to visualize, but imagine a hyper-cube).

They found that the "Volume Growth" ($plov$) cannot be just any number. It has to be a specific value, and there is a forbidden zone in between.

Here is the analogy:
Imagine you are climbing a staircase. You can stand on step 10, or you can jump to step 16. But you cannot stand on steps 11, 12, 13, 14, or 15. There is a "gap" in the stairs.

The Maximal Speed: If the machine is growing as fast as physically possible (the "maximal degree growth"), its volume growth hits a ceiling: $d^2$ (where $d$ is the number of dimensions). For a 4D object, this is $4^2 = 16$ .
The Slower Speeds: If the machine is slightly slower, the volume growth drops significantly. The authors calculated a new, lower ceiling for these slower machines.
The Gap: Between the "slower ceiling" and the "maximal ceiling," there is a wide open space where no machine can exist.

In simple terms:
If you have a 4D object and you find a machine that makes it grow at a certain rate, the authors can tell you immediately: "That rate is impossible." The growth rate must be either very high (near the top) or significantly lower, but it can never be in the middle.

Why is this important?

Rigidity (The "Stiffness" of Math): It shows that these mathematical objects are much more rigid than we thought. You can't just tweak the growth rate slightly; the universe of these shapes forces them into specific, discrete categories.
Solving the "Missing Numbers": Before this, mathematicians knew the top speed and had a guess for the lower speeds, but they didn't know if there were holes in the middle. This paper proves those holes are real.
Connecting to Other Fields: The paper mentions that this growth rate is mathematically identical to a concept in "non-commutative algebra" (a type of abstract algebra used in quantum physics and computer science). So, by understanding the geometry of these shapes, they are also solving puzzles in algebra that have been stuck for decades.

The "Dimension 4" Victory

The paper specifically solves the puzzle for 4-dimensional objects.

If the machine is totally still ( $k=0$ ), the growth is 4.
If it's slightly moving ( $k=2$ ), the growth is 6 or 8.
If it's moving faster ( $k=4$ ), the growth is 10.
If it's at max speed ( $k=6$ ), the growth is 16.

Notice the gap? There are no machines with a growth rate of 11, 12, 13, 14, or 15. The authors proved this gap exists and explained exactly why.

Summary

Think of this paper as a map for a mysterious landscape. For a long time, explorers knew the highest mountain peak and the lowest valley, but they weren't sure if there were cliffs or plateaus in between. Hu and Jiang have drawn a new map showing that there is a massive, empty canyon in the middle. You can't build a house there; the laws of mathematics simply don't allow it.

This "Gap Principle" reveals a hidden order in the chaotic world of geometric transformations, showing that even in high-dimensional space, nature (or math) prefers to be either very fast or significantly slower, with nothing in between.

1. Problem Statement

The paper investigates the polynomial volume growth ( $\text{plov}(f)$ ) of zero-entropy automorphisms $f$ on a normal projective variety $X$ of dimension $d \geq 4$ .

Context: For an automorphism $f$ , the degree growth rate $k$ is defined by $\deg_1(f^n) \sim n^k$ . Zero entropy implies that all dynamical degrees $\lambda_i(f) = 1$ , which forces $k$ to be an even integer satisfying $0 \leq k \leq 2d-2$ .
The Invariant: The polynomial volume growth $\text{plov}(f)$ is defined as the growth rate of the volume of the $n$ -th iterate graph $\Gamma_n \subset X^n$ . Specifically, $\text{plov}(f) = \limsup_{n \to \infty} \frac{\log \text{Vol}(\Gamma_n)}{\log n}$ .
The Gap: Previous work established an upper bound $\text{plov}(f) \leq d^2$ (when $k$ is maximal) and a parity principle ( $\text{plov}(f) \equiv d \pmod 2$ ). However, the precise spectrum of possible values for $\text{plov}(f)$ for non-maximal $k$ was unknown. The authors aim to determine if there are "gaps" in the possible values of $\text{plov}(f)$ and to establish a precise relationship between the degree growth $k$ and $\text{plov}(f)$ .

2. Methodology

The authors employ a sophisticated blend of algebraic geometry, combinatorics, and linear algebra.

A. Algebraic Geometry & Intersection Theory

Formal Log-Monodromy: Instead of using the nilpotent operator $N$ (where $f^* = \text{id} + N$ ), the authors introduce a "formal log-monodromy" operator $L = \log(f^*)$ . This allows them to express the volume $\text{Vol}(\Gamma_n)$ as a polynomial in $n$ with coefficients involving intersection numbers of the form $w_\lambda = (L^{\lambda_1}H_X) \cdots (L^{\lambda_d}H_X)$ .
Vanishing Theorems: They utilize geometric vanishing results (Proposition 3.7) which state that certain intersection numbers $w_\lambda$ vanish if the partition $\lambda$ is "too large" (lexicographically greater than a distinguished partition $\kappa$ ).
Positivity: They establish that specific intersection numbers corresponding to the "distinguished" partition are strictly positive.

B. Combinatorics: Restricted Partitions

The problem is reduced to analyzing restricted partitions $\lambda = (\lambda_1, \dots, \lambda_d)$ where $\sum \lambda_i = n$ and $0 \leq \lambda_i \leq k$ .
The intersection numbers $w_\lambda$ satisfy a system of homogeneous linear equations encoded by weighted incidence matrices $A_{k,d,n}$ .

C. Linear Algebra: Truncated Matrices

Weighted Incidence Matrices: The authors define matrices $A_{k,d,n}$ where entries count the ways to transition between partitions.
Truncation: By combining the geometric vanishing (Proposition 3.7) with the linear system, they define truncated matrices $A^{\leq \kappa}_{k,d,n}$ , which only include columns corresponding to partitions $\lambda \leq \kappa$ .
Nullspace Analysis: The core technical achievement is proving that for specific parameters, these truncated matrices have trivial nullspaces (or one-dimensional nullspaces spanned by a specific vector related to the Vandermonde determinant). This determines whether the leading coefficient of the volume polynomial vanishes.

3. Key Contributions and Results

A. The Main Gap Principle (Theorem 1.1 & 1.2)

The authors establish a sharp dichotomy for $d \geq 4$ :

Maximal Case ( $k = 2d-2$ ): If the degree growth is maximal, then $\text{plov}(f) = d^2$ .
Non-Maximal Case ( $k \leq 2d-4$ ): If the degree growth is not maximal, then $\text{plov}(f) \leq d(d-2) + 2\lfloor d/4 \rfloor$ .

Consequently, for any fixed $d \geq 4$ , the value of $\text{plov}(f)$ cannot lie in the open interval:
$\left( d(d-2) + 2\lfloor d/4 \rfloor, \, d^2 \right)$
This confirms a "gap" in the spectrum of polynomial volume growth.

B. Dimension 4 Classification (Corollary 1.7)

As a direct application, the authors completely classify the possible values of $\text{plov}(f)$ for $d=4$ :

If $k=0$ , $\text{plov}(f) = 4$ .
If $k=2$ , $\text{plov}(f) \in \{6, 8\}$ .
If $k=4$ , $\text{plov}(f) = 10$ .
If $k=6$ (maximal), $\text{plov}(f) = 16$ .
All these values are realizable by 4-dimensional abelian varieties.

C. Connection to Gelfand–Kirillov Dimension

The paper notes that $\text{plov}(f)$ is equal to the Gelfand–Kirillov dimension of the associated twisted homogeneous coordinate ring minus one. Thus, their results provide a new gap principle for the Gelfand–Kirillov dimension of these noncommutative algebras.

4. Technical Highlights

Vandermonde Connection: In the maximal case ( $k=2d-2$ ), the nullspace of the truncated matrix is shown to be one-dimensional, spanned by the coefficient vector of the square of the Vandermonde determinant expressed in normalized monomial symmetric functions. This precise characterization allows the authors to prove the positivity of the leading coefficient, establishing $\text{plov}(f) = d^2$ .
Inductive Nullity Proofs: For the non-maximal case, the authors prove by induction that the truncated matrices have trivial nullspaces for sufficiently large $n$ , forcing the intersection numbers to vanish and thereby lowering the degree of the volume polynomial.
Refined Upper Bounds: They improve the previous general upper bound from $d(d-1)$ to $d(d-2) + 2\lfloor d/4 \rfloor$ for non-maximal cases.

5. Significance

Rigidity Phenomenon: The paper reveals a new rigidity in the dynamics of zero-entropy automorphisms. The polynomial volume growth is not a continuous spectrum but exhibits discrete gaps, particularly separating the maximal growth case from all sub-maximal cases.
Extension of Previous Work: This work extends the results of Artin–Van den Bergh (surfaces, $d=2$ ) and Lin–Oguiso–Zhang (threefolds, $d=3$ ) to arbitrary dimensions $d \geq 4$ .
Conjectures: The authors propose a "Second Gap Principle" (Conjecture 1.8) suggesting that for $d \geq 2$ , $\text{plov}(f)$ cannot lie in $( (d-1)^2+1, d^2 )$ , and a lower bound conjecture (Conjecture 1.9).
Methodological Advance: The introduction of the "formal log-monodromy" operator $L$ and the subsequent analysis of truncated incidence matrices provides a powerful new toolkit for studying dynamical invariants in algebraic geometry.

In summary, Hu and Jiang resolve the precise relationship between degree growth and volume growth for zero-entropy automorphisms in high dimensions, proving that the volume growth jumps discontinuously from a lower bound to the maximum $d^2$ only when the degree growth reaches its theoretical maximum.