On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties

This paper refutes the conjecture that isolated periodic points of automorphisms on affine spaces have bounded height by providing a counterexample, while simultaneously proving that cohomologically hyperbolic dominant rational self-maps on projective varieties possess a Zariski open subset with height-bounded periodic points and offering evidence that such boundedness may fail for preperiodic points.

The Gist

Imagine you are playing a complex video game where you have a character moving around on a grid. Every time you press a button (apply a rule), the character jumps to a new spot based on their current coordinates.

In mathematics, this is called a dynamical system. Mathematicians are very interested in two types of special spots on this grid:

1. Periodic Points: Spots where, if you land there, you eventually loop back to the exact same spot after a few moves. (Like a hamster running in a wheel).
2. Preperiodic Points: Spots where you might wander around for a bit, but eventually, you get sucked into one of those loops. (Like a ball rolling down a hill until it hits a circular track).

The big question this paper asks is: "How far away from the center of the map can these special spots be?"

In math, "how far" is measured by something called Height. Think of Height as the "complexity" or "size" of the numbers needed to describe the spot. A spot at $(1, 1)$ has low height. A spot at $(1,000,000, 1,000,000)$ has high height.

The Big Mystery (The Conjecture)

For a long time, mathematicians believed a rule called Conjecture 1.1. They thought:

"If you have a game with a complex rule (degree $\ge$ 2), all the 'looping' spots (periodic points) must stay within a certain distance from the center. You can't have loops that get infinitely far away."

They thought the "looping" spots were like a flock of birds that, no matter how long they fly, never stray beyond a specific horizon.

The Plot Twist: The Counterexample (Section 2)

The authors, Matsuzawa and Sano, say: "Not so fast!"

They built a specific 3-dimensional game (a map on a 3D space) with a very tricky rule.

• The Trick: They designed the game so that the first two coordinates act like a known chaotic system (a Henon map), but the third coordinate acts like a "tug-of-war" rope.
• The Result: They found an infinite sequence of looping spots. As you go further out in the sequence, the numbers describing these spots get massively larger and larger.
• The Analogy: Imagine a spiral staircase that goes up forever. The authors found a way to build a staircase where the steps (the periodic points) keep getting higher and higher, defying the rule that said "there must be a ceiling."

They proved that for this specific 3D game, the "height" of the periodic points is unbounded. The birds can fly as far as they want.

The Silver Lining: When the Rule Does Hold (Section 3)

Just because the rule failed for one specific game doesn't mean it's useless. The authors asked: "Under what conditions does the rule actually work?"

They found a special category of games called Cohomologically Hyperbolic Maps.

• The Metaphor: Imagine a game where the rules stretch space in one direction and shrink it in another, like a hyperbolic saddle. The "chaos" is very organized.
• The Discovery: For these "well-behaved" chaotic games, the rule IS TRUE. If you look at a large enough open area of the map, all the looping spots inside that area are bounded. They stay within a safe distance.
• The Takeaway: The universe of math is consistent, but you have to pick the right kind of game to see the pattern. If the game is "hyperbolic" (organized chaos), the birds stay in the flock.

The Final Twist: The "Almost" Loops (Section 4)

The authors then looked at Preperiodic points (the ones that wander before looping).

• The Question: Even in those "well-behaved" hyperbolic games, do the wandering spots stay bounded?
• The Result: They built another example suggesting the answer is NO.
• The Analogy: Imagine a ball rolling down a hill. Even if the hill is perfectly shaped (hyperbolic), the ball might roll down a very long, winding path before it hits the circular track. The authors showed you can construct a path so long and complex that the ball's "height" (complexity) becomes infinite before it finally settles into a loop.

Summary for the General Audience

1. The Old Belief: Mathematicians thought all "looping" spots in complex systems had to stay close to home.
2. The Breakthrough: The authors found a 3D system where these spots fly off to infinity, breaking the old belief.
3. The Correction: However, they proved that for a specific, well-structured type of chaos (hyperbolic maps), the spots do stay bounded, provided you look at the right part of the map.
4. The Warning: Even in those well-structured systems, if you look at points that almost loop (preperiodic), they might still fly off to infinity.

In short: The universe of these mathematical games is wilder than we thought. Sometimes the loops go to infinity, sometimes they stay close, and sometimes the path to the loop is infinitely long. The authors gave us the map to tell the difference.

Technical Summary

Here is a detailed technical summary of the paper "On the Height Boundedness of Periodic and Preperiodic Points of Dominant Rational Self-Maps on Projective Varieties" by Yohsuke Matsuzawa and Kaoru Sano.

1. Problem Statement and Context

The paper addresses a central question in arithmetic dynamics: Under what conditions is the set of (pre)periodic points of a self-map on an algebraic variety height-bounded?

• Background: For a non-isomorphic surjective morphism $f: \mathbb{P}^N_{\mathbb{Q}} \to \mathbb{P}^N_{\mathbb{Q}}$, the set of preperiodic points is known to be height-bounded (due to the Northcott property of the canonical height function).
• The Conjecture: A specific conjecture (Conjecture 1.1, attributed to Silverman) posits that for an automorphism $f: \mathbb{A}^N_{\mathbb{Q}} \to \mathbb{A}^N_{\mathbb{Q}}$ where the maximum degree of coordinate functions is at least two, the set of isolated periodic points should be height-bounded.
• The Gap: While this conjecture holds for dimensions $N=2$ and for specific classes of automorphisms (regular affine, triangular) in higher dimensions, the general case for $N \ge 3$ was open. Furthermore, the behavior of preperiodic points for more general rational maps (specifically cohomologically hyperbolic ones) was not fully understood.

2. Methodology

The authors employ a combination of arithmetic geometry, dynamical systems theory, and algebraic geometry techniques:

• Counterexample Construction (Section 2):

  • They construct a specific automorphism on $\mathbb{A}^3$ by coupling a Hénon map on $\mathbb{A}^2$ with a linear recurrence in the third coordinate.
  • They utilize reduction modulo primes (specifically $p=3$) to analyze the structure of periodic points.
  • They apply the Jacobian criterion and properties of étale morphisms to study the scheme of periodic points over $\mathbb{Z}$.
  • They use trace maps over finite fields to estimate the $p$-adic valuation of the coordinates, demonstrating that the height grows unboundedly.

• Positive Results via Cohomological Hyperbolicity (Section 3):

  • They utilize the concept of cohomological hyperbolicity, defined by the condition that a unique dynamical degree $d_p(f)$ is strictly maximal among all dynamical degrees $d_i(f)$.
  • The proof relies on recursive height inequalities derived from the pullback of ample divisors.
  • They use the theory of big $\mathbb{R}$-divisors and birational models to establish lower bounds on height functions along orbits, proving that periodic points must lie within a bounded region if the map is cohomologically hyperbolic.

• Preperiodic Counterexamples (Section 4):

  • They construct a family of rational maps on $\mathbb{P}^2$ with specific dynamical degrees ($d_1 > d_2$).
  • They analyze backward orbits of fixed points. By studying the growth of the degree of the number fields generated by preimages and the $p$-adic valuations of coordinates, they show that preperiodic points can have unbounded height even when the map is cohomologically hyperbolic.

3. Key Contributions and Results

A. Disproof of the Conjecture for $\mathbb{A}^3$ (Theorem 1.3 & 2.1)

The authors provide a counterexample to Conjecture 1.1 in dimension 3.

• The Map: $f: \mathbb{A}^3_{\mathbb{Q}} \to \mathbb{A}^3_{\mathbb{Q}}$ defined by $(x, y, z) \mapsto (y, x + y - y^3, 2z - y)$.
• Properties:
  1. $f$ is an automorphism.
  2. The set of $n$-periodic points is finite for all $n$ (all periodic points are isolated).
  3. The dynamical degrees are $d_1(f) = d_2(f) = 3$.
• Result: There exists a sequence of periodic points $\{P_i\}$ that is Zariski dense in $\mathbb{A}^3$ such that $\lim_{i \to \infty} h(P_i) = \infty$.
• Significance: This shows that the "isolated periodic points" conjecture fails in dimension 3, primarily because the map is not cohomologically hyperbolic (the first two dynamical degrees are equal).

B. Height Boundedness for Cohomologically Hyperbolic Maps (Theorem 1.8)

The authors prove a positive result for a broader class of maps.

• Theorem: If $f: X \dashrightarrow X$ is a cohomologically hyperbolic dominant rational self-map on a projective variety over $\mathbb{Q}$, then there exists a non-empty Zariski open subset $U \subset X$ such that the set of periodic points $P$ with $O_f(P) \subset U$ is height-bounded.
• Refinement: This result generalizes previous work by Wang and Matsuzawa-Wang by removing the assumption that $f$ must be birational.
• Uniformity: Theorem 3.1 extends this to families of maps, providing a uniform upper bound on the height of periodic points in terms of the height of the parameter of the family.

C. Failure of Boundedness for Preperiodic Points (Theorem 1.9)

The authors investigate whether the boundedness result holds for preperiodic points.

• The Map: A family of rational maps on $\mathbb{P}^2$ (e.g., for $d=4, p=2$) that is 1-cohomologically hyperbolic ($d_1=4, d_2=2$).
• Result: They construct a sequence of preperiodic points $\{P_n\}$ (specifically, a backward orbit of a fixed point) such that:
  1. The set $\{P_n\}$ is Zariski dense.
  2. $\lim_{n \to \infty} h(P_n) = \infty$.
• Significance: This suggests that the condition "cohomologically hyperbolic" is not sufficient to guarantee height boundedness for preperiodic points, unlike for periodic points. The condition that $f$ is $dim(X)$-cohomologically hyperbolic (as in Theorem 1.6(2)) may be essential.

4. Significance and Future Directions

• Resolution of Conjecture 1.1: The paper definitively settles the conjecture for automorphisms of affine space, showing it is false in dimension 3. It highlights that the failure is linked to the lack of cohomological hyperbolicity (specifically, the equality of dynamical degrees).
• Role of Cohomological Hyperbolicity: The work establishes cohomological hyperbolicity as the correct geometric condition for ensuring height boundedness of periodic points on projective varieties.
• Distinction between Periodic and Preperiodic: A major contribution is the demonstration that the arithmetic behavior of preperiodic points is fundamentally different from periodic points. Even for "hyperbolic" maps, preperiodic points can escape height bounds, challenging the expectation of a unified Northcott-type property for all (pre)periodic points.
• Open Questions: The authors leave open the question of whether there exists a cohomologically hyperbolic map where the set of preperiodic points is never height-bounded on any Zariski open set (Question 1.10). They suspect their example in Theorem 1.9 might be such a case, though proving the points cannot be defined over a single number field remains a technical hurdle.

In summary, this paper refines the landscape of arithmetic dynamics by delineating the precise boundary between maps that exhibit Northcott-like finiteness/height-boundedness properties and those that do not, distinguishing sharply between the behaviors of periodic and preperiodic points.