$\log$-Hölder regularity of currents and equidistribution towards Green currents

This paper establishes that for endomorphisms of projective spaces or automorphisms of compact Kähler manifolds, the pull-backs of currents under iterates converge exponentially fast to Green currents when tested against $\log$-Hölder continuous observables with bounded $\mathrm{dd^c}$ mass.

The Gist

Here is an explanation of Marco Vergamini's paper, translated into simple language with creative analogies.

The Big Picture: The "Cosmic Blender"

Imagine you have a magical blender (a mathematical system) that takes a shape, chops it up, mixes it, and spits it back out. If you do this over and over again, the original shape gets lost. Eventually, no matter what you put in, the blender produces the same "perfectly mixed" smoothie.

In mathematics, this "perfectly mixed" state is called a Green Current. It represents the ultimate equilibrium of a complex system.

The paper asks a very specific question: How fast does the blender mix things up? And more importantly, does it mix things up evenly even if the ingredients we put in are a bit "rough" or "jagged"?

The Ingredients: Smooth vs. Rough

In the world of complex geometry (the study of shapes in higher dimensions), we usually test our blenders with smooth ingredients (like silk or polished glass). Mathematicians have known for a long time that if you use smooth ingredients, the blender mixes them perfectly, and the speed of mixing is exponential (it happens very fast).

However, real-world data (or "observables" in math terms) isn't always smooth. Sometimes it's a bit bumpy, like sandpaper or a crumpled piece of paper.

• Hölder Continuous: Think of this as "smooth but slightly bumpy." If you rub your finger over it, it feels okay, but not perfect.
• Log-Hölder Continuous: This is the paper's star ingredient. It's like very rough sandpaper. It's much "rougher" than the smooth type. In fact, it's so rough that standard mathematical tools usually break when they try to touch it.

The Problem: Previous studies showed that if you use "rough" ingredients, the blender might mix them slowly, or the math might break entirely.

The Breakthrough: Vergamini proves that even with these "rough" (Log-Hölder) ingredients, the blender still mixes them exponentially fast. It's as if the blender is so powerful that it doesn't care if the ingredients are jagged; it smoothes them out just as quickly as it would smooth out silk.

The Two Main Scenarios

The paper looks at two types of blenders:

1. The Reversible Blender (Automorphisms of Kähler Manifolds):

   • Analogy: Imagine a machine that can run forward and backward perfectly. If you mix the ingredients, you can un-mix them exactly.
   • The Result: Vergamini shows that even with rough ingredients, the mixing towards the "perfect smoothie" happens at a predictable, lightning-fast speed.

2. The One-Way Blender (Endomorphisms of Projective Spaces):

   • Analogy: Imagine a machine that only runs forward. Once you mix the ingredients, you can't un-mix them. This is harder to analyze because information gets lost (like shredding a document).
   • The Result: This is the harder part of the paper. Because the machine destroys information, the "roughness" of the ingredients usually gets worse as they go through the machine. Vergamini had to invent a new way of measuring "roughness" (using something called Super-Potentials) to prove that even in this one-way machine, the mixing is still fast and reliable.

The Secret Weapon: "Super-Potentials"

To measure how "rough" an ingredient is, the author uses a tool called a Super-Potential.

• Analogy: Imagine you want to measure the roughness of a mountain range. You can't just look at the rocks; you need to look at the shadow the mountain casts.
• In this paper, the "shadow" (the Super-Potential) tells the mathematician how the current (the ingredient) behaves. Vergamini discovered that even if the ingredient is "Log-Hölder" (very rough), its "shadow" behaves in a very controlled, predictable way. This allowed him to prove the mixing speed remains fast.

Why Does This Matter? (The "So What?")

You might wonder, "Who cares if a mathematical blender mixes sandpaper fast?"

1. Predicting Chaos: In the real world, many systems are chaotic (weather, stock markets, fluid dynamics). This paper helps us understand how quickly these systems settle into a predictable pattern, even if the starting data is messy.
2. Statistics of Chaos: The paper mentions "Mixing" and "Central Limit Theorem." In plain English, this means we can now predict the average behavior of these chaotic systems with much higher precision, even when the inputs are imperfect.
3. Geometry of the Universe: It helps mathematicians understand the hidden geometric structures of complex spaces, proving that certain "rough" features eventually smooth out into a perfect, stable shape.

Summary in One Sentence

Marco Vergamini proved that even if you feed "rough, jagged" data into complex mathematical machines, they still settle into a perfect, stable state just as fast as they do with smooth data, thanks to a new way of measuring the "roughness" of the input.

Technical Summary

Here is a detailed technical summary of the paper "Log-Hölder Regularity of Currents and Equidistribution Towards Green Currents" by Marco Vergamini.

1. Problem Statement and Context

The paper addresses the quantitative rate of equidistribution in holomorphic dynamical systems. Specifically, it investigates how the pull-backs of currents under the iterates of a map $f$ converge to the Green currents (invariant currents of maximal dynamical degree).

• Classical Setting: Previous results (e.g., by Dinh and Sibony) established exponential convergence rates when testing currents against Hölder-continuous observables (forms or functions).
• The Gap: Hölder continuity is a strong condition. In many dynamical contexts, observables may only satisfy a weaker log-Hölder (or log-continuous) condition, defined by a modulus of continuity $w(\delta) \sim (\log \delta)^{-q}$.
• The Challenge: While log-Hölder regularity is preserved under push-forwards of holomorphic maps (unlike standard Hölder regularity, which often degrades), extending this regularity to currents of arbitrary bidegree is non-trivial. In higher bidegrees, the push-forward of a smooth form is not necessarily continuous, making standard function-based regularity theories insufficient.
• Objective: To prove that equidistribution towards Green currents occurs at an exponential rate when tested against log-Hölder-continuous forms (with bounded mass of their $dd^c$) for both:
  1. Automorphisms of compact Kähler manifolds.
  2. Endomorphisms of projective spaces ($\mathbb{P}^k$).

2. Methodology

The author employs the theory of super-potentials developed by Dinh and Sibony, shifting the focus from the regularity of the test forms themselves to the regularity of the super-potentials of the currents.

Key Technical Tools:

1. Log-Hölder Continuous Super-Potentials:

   • The paper defines a current $R$ to have a log-Hölder-continuous super-potential if its super-potential functional $U_R$ satisfies a modulus of continuity $w(\delta) = M(1+|\log \delta|)^{-q}$.
   • This is a weaker condition than Hölder continuity but sufficient for the dynamical applications.

2. Skoda-Type Estimates for Currents:

   • A central technical contribution is Proposition 3.19, a "Skoda-type" estimate. It bounds the value of a super-potential $U_S(R)$ where $R$ has a log-Hölder continuous super-potential.
   • The estimate takes the form: $|U_S(R)| \lesssim 1 + M^{\frac{1}{q+1}}$. This allows the control of the interaction between currents with different regularity levels.

3. Preservation of Regularity under Iteration:

   • Automorphisms (Invertible Case): The paper proves that the pull-back operator $f^*$ preserves the log-Hölder continuity of super-potentials with controlled growth of the multiplicative constant (Proposition 4.2).
   • Endomorphisms (Non-Invertible Case): This is the most delicate part. Since $f$ is not invertible, the push-forward $f_*$ of a smooth form may not be continuous. The author introduces a specific operator $L_\theta$ (structural line) and proves that the pull-back operator $L_{k-s}$ is Hölder continuous with respect to the distance $\text{dist}_2$ (Lemma 5.2). This allows the propagation of log-Hölder regularity through iterations despite the non-invertibility.

4. Duality Argument:

   • The convergence of currents is analyzed by duality. Instead of studying the convergence of the current directly, the paper studies the convergence of the super-potentials evaluated at the test forms.

3. Key Contributions

• Definition of Log-Hölder Regularity for Currents: The paper successfully extends the concept of log-Hölder regularity from functions/forms to the space of currents via super-potentials.
• Quantitative Equidistribution for Log-Hölder Observables: It establishes that the convergence rate remains exponential even when the test functions are only log-Hölder continuous, provided their $dd^c$ has bounded mass.
• Handling Non-Invertible Maps: The paper overcomes the difficulty of non-invertibility in projective spaces by proving that the pull-back operator acts as a contraction on the space of currents with specific regularity, utilizing a precise control of the multiplicative constants involving the degree $d$ and the dimension $k$.
• Statistical Applications: The results are applied to derive exponential mixing of all orders and the Central Limit Theorem (CLT) for log-Hölder observables in the context of Kähler automorphisms.

4. Main Results

Theorem 1.1 (Automorphisms of Compact Kähler Manifolds)

Let $f$ be an automorphism of a compact Kähler manifold $X$ with simple action on cohomology. Let $T_+$ be the Green current and $d_p$ the main dynamical degree.
For any log-Hölder continuous form $\Phi$ (with bounded $dd^c$ mass) and any current $S$ of mass 1:
$$\left| \left\langle \frac{(f^*)^n}{d_p^n}(S) - r T_+, \Phi \right\rangle \right| \lesssim \|\Phi\|_q \left( \frac{\delta}{d_p} \right)^{\frac{nq}{q+1}}$$
where $\delta < d_p$ is an auxiliary dynamical degree, $r$ is the limit constant, and the implicit constant depends on $f, q$.

Theorem 1.2 (Endomorphisms of Projective Spaces)

Let $f$ be a generic endomorphism of $\mathbb{P}^k$ of degree $d$.
For any bidegree $(s, s)$, any log-Hölder continuous form $\Phi$, and any positive closed current $S$ of mass 1:
$$\left| \left\langle \frac{(f^*)^n}{d^n}(S) - T^s, \Phi \right\rangle \right| \lesssim \|\Phi\|_q \left( \frac{\kappa}{d} \right)^{\frac{nq}{q+1}}$$
where $T^s$ is the Green current, and $\kappa < d$ is a constant related to the geometry of the map (specifically involving the maximal multiplicity $m$ and dimension $k$).

Theorem 4.5 (Exponential Mixing)

The equilibrium measure $\mu = T_+ \wedge T_-$ is exponentially mixing of all orders for observables bounded in the log-Hölder norm $\|\cdot\|_q$. This implies the Central Limit Theorem for these observables.

5. Significance and Impact

• Optimality of Rates: The paper demonstrates that log-Hölder regularity is a "natural" class for holomorphic dynamics because, unlike standard Hölder regularity, the exponent of regularity does not degrade under the dynamics. This allows for sharper statistical results.
• Extension to Higher Bidegrees: By moving the regularity condition to the super-potential, the author bypasses the issue that push-forwards of smooth forms in higher bidegrees are not continuous. This generalizes previous results that were often restricted to $(1,1)$-currents or specific bidegrees.
• Statistical Mechanics: The results provide a robust framework for deriving statistical properties (mixing, CLT) for a broader class of observables than previously possible, bridging the gap between smooth and merely continuous dynamics.
• Future Directions: The author notes that these techniques are adaptable to holomorphic correspondences (addressed in a companion work) and suggests potential extensions to horizontal-like maps, though non-compactness poses new challenges.

In summary, Vergamini's work refines the understanding of convergence rates in complex dynamics by introducing a robust regularity framework (log-Hölder super-potentials) that is stable under iteration, thereby extending exponential equidistribution and statistical limit theorems to a wider class of dynamical systems and observables.