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On the dynamical degree of surjective endomorphisms

This paper establishes dynamical properties of surjective rational maps on smooth projective surfaces, provides a numerical characterization of regular maps on del Pezzo surfaces, and presents explicit constructions related to topological entropy.

Original authors: Ilya Karzhemanov

Published 2026-03-26

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Ilya Karzhemanov

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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, infinitely stretchable piece of fabric (this is your surface, like a sphere or a torus, but in the world of complex geometry). Now, imagine you have a special tool, a map (let's call it $f$ ), that takes this fabric, crumples it, stretches it, and lays it back down on itself.

Sometimes, this tool works perfectly: it covers the whole fabric without tearing or leaving holes. This is a regular map. Other times, the tool is a bit glitchy; it tries to cover the fabric but leaves a few tiny, messy spots where the math gets undefined (like a pixelated glitch in a video game). This is a rational map.

The paper by Ilya Karzhemanov is like a detective story trying to figure out: "Is this map a perfect, smooth operator, or is it glitchy?" And more importantly, "How chaotic does the fabric get when we keep using this tool over and over?"

Here is the breakdown of the paper's main ideas using everyday analogies:

1. The "Growth Meter" (The Dynamical Degree)

When you use the map $f$ once, the fabric might stretch a little. If you use it twice ( $f$ then $f$ again), it stretches more. If you use it a million times, the fabric might be stretched into a giant, tangled mess.

The author introduces a number called the Dynamical Degree ( $\lambda$ ). Think of this as the "Speedometer of Chaos."

It measures how fast the "complexity" of the fabric grows with every use of the map.
If $\lambda$ is high, the fabric is getting wildly complicated very fast.
If $\lambda$ is low, the fabric stays relatively tame.

The Big Discovery: The author proves that for these specific types of surfaces (called del Pezzo surfaces, which are like the "perfectly shaped" geometric spheres), this Speedometer ( $\lambda$ ) is always a nice, clean number (an algebraic integer). It's not some weird, unmeasurable decimal like $\pi$ . It's a number you can write down with a ruler and a compass.

2. The "Smoothness Test" (Regular vs. Glitchy)

The most exciting part of the paper is a litmus test to see if the map is "regular" (perfect) or "glitchy" (rational).

The author compares two numbers:

The Speedometer ( $\lambda$ ): How fast the complexity grows.
The "Degree" ( $\deg f$ ): How many times the map wraps the fabric around itself (like how many times you wrap a scarf around a pole).

The Golden Rule:

If you square the Speedometer ( $\lambda^2$ ) and it equals the Degree ( $\deg f$ ), then the map is perfectly smooth. No glitches.
If $\lambda^2$ is bigger than the Degree, then the map is glitchy. It has those messy spots where the math breaks down.

The Analogy:
Imagine a photocopier.

Perfect Map: You put a photo in, and it comes out sharp. The "sharpness growth" matches the "paper usage" exactly.
Glitchy Map: You put a photo in, and it comes out blurry or pixelated. The "blur growth" is higher than the "paper usage" would suggest. The paper is being wasted on the glitches.

The author proves that for these specific surfaces, if the numbers match, the machine is working perfectly. If they don't, the machine is broken.

3. The "Rank 1" Mystery (The One-Way Street)

The paper also looks at a special class of maps called "Rank 1" maps.
Imagine a map that squashes a 3D object down into a 2D line, and then projects it back. It's a very specific, "degenerate" way of moving things.

The author shows that even though these maps seem simple (they only have one "direction" of movement), they can still be surprisingly tricky.

The Surprise: You might think, "Oh, this map is just a projection from a flat sheet (like a shadow)."
The Reality: The author constructs a family of these maps and proves that for most of them, you cannot simply describe them as a shadow of a flat sheet. They have a hidden, twisted structure that defies simple explanation.

It's like looking at a shadow of a complex sculpture and thinking, "That's just a flat circle," but then realizing the object casting the shadow is actually a twisted knot that you can't flatten out.

4. Why Does This Matter? (Entropy)

Finally, the paper touches on Entropy. In simple terms, entropy is a measure of disorder.

If you keep using the map, does the fabric become a predictable, orderly pattern?
Or does it become a chaotic, unpredictable mess?

The author calculates that for these maps, the chaos is bounded. You can predict the maximum amount of disorder the fabric can ever reach. This is useful for physicists and computer scientists who study how systems evolve over time, because it tells them that even in complex geometric worlds, there are limits to how chaotic things can get.

Summary in a Nutshell

Ilya Karzhemanov is a detective looking at a specific type of geometric surface. He found a simple math equation ( $\lambda^2 = \deg f$ ) that acts as a truth detector:

If the equation holds: The map is smooth and perfect.
If the equation fails: The map has glitches.

He also showed that even the "simplest" looking maps can have hidden, twisted structures that aren't what they seem. It's a reminder that in the world of geometry, things are often more complex and interesting than they appear on the surface.

1. Problem Statement and Context

The paper investigates the dynamical properties of surjective rational maps $f: X \dashrightarrow X$ on smooth projective complex surfaces $X$ , with a specific focus on del Pezzo surfaces (where the anticanonical divisor $-K_X$ is ample).

The central objects of study are:

The Dynamical Degree ( $\lambda(f)$ ): A numerical invariant measuring the asymptotic growth rate of the action of $f$ on the Neron-Severi group $N^1(X) = \text{Pic}(X) \otimes \mathbb{R}$ . It is defined as $\lambda(f) = \lim_{n \to \infty} \sqrt[n]{(f^n)^*(H) \cdot H}$ for an ample divisor $H$ .
The Topological Degree ( $\deg f$ ): The degree of the map as a covering map (where defined).
Regularity: Whether the map $f$ is a morphism (regular everywhere) or merely a rational map with an indeterminacy locus.

The paper aims to establish the relationship between $\lambda(f)$ and $\deg f$ , specifically characterizing when a surjective rational map is actually a regular morphism, and exploring the topological entropy of such maps.

2. Methodology

The author employs a blend of algebraic geometry and linear spectral theory:

Spectral Interpretation (Krein-Rutman Theorem): The paper utilizes the fact that the pullback operator $f^*: N^1(X) \to N^1(X)$ preserves the nef cone (the cone of numerically effective classes). By applying the Krein-Rutman theorem (a generalization of the Perron-Frobenius theorem for cones), the author establishes the existence of a nef eigenvector $v$ corresponding to the spectral radius $\rho(f^*)$ .
Resolution of Indeterminacy: To relate $\lambda(f)$ and $\deg f$ , the author considers a resolution diagram $Y \xrightarrow{g} X \xrightarrow{h} X$ , where $g$ is a composition of blow-ups and $h$ is a regular morphism. This allows the decomposition of the pullback of divisors into a regular part and an exceptional part ( $E$ ).
Intersection Theory and Hodge Index Theorem: The proofs rely heavily on intersection numbers of divisors. The Hodge Index Theorem is used to establish inequalities between intersection products, specifically $(f^*(H) \cdot H)^2 \geq (f^*(H)^2)(H^2)$ .
Deformation Arguments: In the final section, the author uses flat families of rational maps to demonstrate that certain structural properties (like factoring through $\mathbb{P}^2$ ) are not generic, even for simple surfaces like $\mathbb{P}^1 \times \mathbb{P}^1$ .

3. Key Contributions and Results

A. Spectral Characterization of the Dynamical Degree

The paper provides a transparent proof of a classical result (Proposition 1.2):

Equality: The dynamical degree equals the spectral radius of the pullback operator: $\lambda(f) = \rho(f^*)$ .
Existence of Eigenvector: There exists a nef class $v \in N^1(X)$ such that $f^*(v) = \lambda(f)v$ .
Algebraic Nature: Since $f^*$ preserves the lattice $\text{Pic}(X)$ , $\lambda(f)$ is an algebraic integer for surjective maps on surfaces with irregularity $q(X)=0$ . (This contrasts with general dominant rational maps where $\lambda(f)$ can be transcendental).

B. The Inequality $\lambda(f)^2 \geq \deg f$

The author proves a log-concavity property for surjective maps on surfaces:
$\lambda(f)^2 \geq \deg f$
This is derived by analyzing the resolution of the map. The proof shows that the "loss" of positivity due to exceptional divisors ( $E$ ) in the pullback formula ensures that the intersection product $(f^*(H)^2)$ is strictly less than or equal to $(\deg f)(H^2)$ unless the map is regular.

C. Numerical Characterization of Regularity (Theorem 1.5)

This is the paper's primary theoretical contribution. The author establishes a necessary and sufficient condition for a surjective rational map on a del Pezzo surface to be regular:

Theorem: A surjective rational map $f: X \dashrightarrow X$ (where $X$ is a del Pezzo surface) is regular if and only if $\lambda(f)^2 = \deg f$ .

Forward Direction: If $f$ is regular, $f^*f_* = (\deg f)\text{Id}$ , making the equality evident.
Reverse Direction: The proof is non-trivial. It assumes $\lambda(f)^2 = \deg f$ and $f$ is non-regular (implying a non-trivial exceptional divisor $E$ ). By testing the equality against the Perron eigenvector $v$ , the author derives a contradiction. Crucially, this relies on the specific geometry of del Pezzo surfaces (rational polyhedral Mori cone) to handle the case where $v^2 = 0$ .

D. Rank 1 Maps and Entropy

The paper introduces and analyzes Rank 1 maps (Definition 4.5), where the operator $f^*$ has rank 1.

For these maps, $\lambda(f)$ is an integer (equal to the trace of $f^*$ ).
The author constructs explicit examples of surjective endomorphisms on $\mathbb{P}^1 \times \mathbb{P}^1$ that are not compositions of a surjective map to $\mathbb{P}^2$ and a projection back (i.e., they do not factor through $\mathbb{P}^2$ ).
Topological Entropy: The paper discusses the upper bound for topological entropy $h(f) \leq \max\{\log \lambda(f), \log \deg f\}$ . Several explicit examples are provided where $\lambda(f) < \deg f$ , illustrating the gap between algebraic and topological complexity.

4. Significance

Clarification of Regularity: The paper resolves a specific question regarding the "algebraic stability" of surjective maps. It provides a purely numerical test ( $\lambda(f)^2 = \deg f$ ) to determine if a rational map is actually a morphism, a distinction that is often difficult to verify directly on the map's definition.
Del Pezzo Specifics: The results highlight the rigidity of del Pezzo surfaces. The proof of the regularity theorem fails for arbitrary rational surfaces (as noted in Remark 1.6 regarding monomial maps on $\mathbb{P}^2$ ), emphasizing the unique geometric constraints of del Pezzo surfaces.
Counter-intuitive Structures: The construction of rank 1 maps that do not factor through $\mathbb{P}^2$ challenges the intuition that "simple" surfaces have "simple" endomorphism structures. It suggests a rich, non-trivial class of surjective rational maps that are not easily reducible to standard projections.
Entropy Bounds: By providing explicit calculations for $\lambda(f)$ and $\deg f$ , the paper offers concrete data points for understanding the upper bounds of topological entropy in algebraic dynamics.

In summary, Karzhemanov's work bridges spectral theory and intersection theory to provide a sharp numerical criterion for the regularity of surjective endomorphisms on del Pezzo surfaces, while simultaneously expanding the known landscape of rational maps through explicit, non-standard constructions.