A Ruelle-McMullen formula for the volume dimension of skew products in $\mathbb C^2$

This paper establishes an explicit second-order expansion for the volume dimension of the Julia sets of holomorphic skew products in $\mathbb{C}^2$ as a perturbation parameter approaches zero, extending the classical Ruelle-McMullen results on Hausdorff dimension to non-conformal higher-dimensional dynamical systems.

The Gist

Imagine you are an architect studying a very strange, magical building. This building isn't made of bricks, but of math. It's a "fractal" structure—a shape that looks infinitely complex no matter how much you zoom in.

In the world of one-dimensional math (like a flat line), mathematicians have long known how to measure the "roughness" or "complexity" of these shapes. They call this the Hausdorff dimension. Think of it like measuring how much paint you need to cover a crinkly piece of paper. If the paper is smooth, you need a little paint (dimension 1). If it's super crinkly, you need a lot more (dimension 1.5, for example).

Back in the 1980s, a mathematician named David Ruelle discovered something fascinating: If you take a simple, perfect shape (like a circle) and wiggle it just a tiny bit, the "roughness" of the resulting shape increases in a very predictable way. It's like saying, "If I push this smooth hill just a little bit, the amount of gravel on its surface will increase by exactly this amount."

Later, Curtis McMullen proved this rule works for many different types of hills.

The Problem: The 3D Twist

The authors of this paper, Fabrizio Bianchi and Yan Mary He, wanted to see if this rule works in a more complex world: two dimensions (like a sheet of paper floating in 3D space, or a "skew product" in $\mathbb{C}^2$).

Here's the catch: In one dimension, the math behaves nicely and uniformly (like a rubber sheet stretching evenly). But in two dimensions, the math is non-conformal. Imagine stretching a rubber sheet where you pull the top hard but the bottom barely moves. The shape distorts in weird, uneven ways. Because of this, the old "paint measurement" (Hausdorff dimension) stops making sense. It's like trying to measure the roughness of a crumpled ball of foil by only looking at its shadow; you lose the 3D information.

The Solution: The "Volume Dimension"

To fix this, the authors introduced a new ruler called the Volume Dimension.

• The Analogy: Imagine you have a sponge. In 1D, you measure how much water it holds by looking at its length. In 2D, you need to know how much water it holds by looking at its volume and how the water flows through its holes.
• The Volume Dimension is a special number that captures the complexity of the shape while respecting how the math stretches and squashes it unevenly. It's a "dynamically defined" ruler, meaning it's built specifically for these moving, shifting shapes.

The Experiment: The Skew Product

The authors studied a specific family of these 2D shapes, which they call Skew Products.

• The Setup: Imagine a machine with two dials, $z$ and $w$.
  • Dial $z$ spins in a circle ($z \to z^d$).
  • Dial $w$ spins based on where $z$ is, plus a little bit of "noise" controlled by a tiny knob $t$.
• The Question: If we turn that tiny knob $t$ just a tiny bit (from 0 to a small number), how much does the "Volume Dimension" of the resulting fractal change?

The Discovery: The "Second-Order" Formula

The authors found a precise formula, similar to Ruelle's and McMullen's, but for this new 2D world.

Here is the simple takeaway:

1. At the start (Knob $t=0$): The shape is perfectly symmetric. Its Volume Dimension is exactly 1.5 (or $1/2$ in their specific scaling, which corresponds to the standard 1D dimension of 1).
2. The Wiggle (Knob $t$ is small): As you turn the knob, the dimension doesn't change linearly (it doesn't just go up a little bit). It stays flat for a split second and then curves upward.
3. The Formula: The increase in complexity depends on the square of how much you turned the knob ($|t|^2$) and the average strength of the "noise" coefficients ($c_k$) in the machine.

The Metaphor:
Think of the fractal as a calm lake.

• Ruelle/McMullen showed that if you drop a pebble in a 1D river, the ripples spread out in a predictable pattern.
• Bianchi and He showed that if you drop a pebble in a 2D lake that has strong currents and eddies (the non-conformal part), the ripples still spread out predictably, but you have to measure the "volume" of the water displaced, not just the width of the ripples.
• Their formula tells you exactly how much "water" (complexity) is displaced based on the size of the pebble and the strength of the currents.

Why Does This Matter?

This is a big deal because:

1. It bridges the gap: It takes a famous, beautiful result from simple math and successfully translates it into the messy, complex world of higher dimensions.
2. It provides a new tool: By defining the "Volume Dimension" and proving it behaves nicely (it's smooth and predictable), they give future mathematicians a reliable way to measure chaos in 2D systems.
3. It confirms stability: Even though these 2D systems are wild and non-conformal, their complexity doesn't explode randomly. It follows a strict, elegant law.

In a nutshell: The authors built a new ruler for 2D fractals, proved that this ruler behaves smoothly when you tweak the system, and wrote down the exact math for how much "roughness" you get for every tiny push you give the system.

Technical Summary

Here is a detailed technical summary of the paper "A Ruelle-McMullen Formula for the Volume Dimension of Skew Products in $\mathbb{C}^2$" by Fabrizio Bianchi and Yan Mary He.

1. Problem Statement and Motivation

Context:
In the 1980s, David Ruelle established that the Hausdorff dimension of the Julia set for the quadratic family $f_c(z) = z^2 + c$ admits a second-order expansion near $c=0$. Specifically, the dimension has a strict local minimum at the monomial map $z \mapsto z^2$. Later, Curtis McMullen generalized this to polynomial perturbations of $z^d$, providing an explicit formula for the second derivative of the Hausdorff dimension.

The Challenge in Higher Dimensions:
The authors aim to extend these results to holomorphic dynamical systems in higher dimensions, specifically skew products in $\mathbb{C}^2$. However, a fundamental obstacle exists:

• Non-conformality: Holomorphic maps in dimensions $>1$ are non-conformal.
• Failure of Hausdorff Dimension: In this setting, the standard Hausdorff dimension is not well-adapted to the dynamics. It lacks a clear relationship with entropy and Lyapunov exponents, and its behavior in parameter space is poorly understood due to the unavailability of tools like Koebe's distortion theorem.

The Solution:
The authors utilize the Volume Dimension ($VD$), a notion they introduced in prior work ([BH24]).

• $VD$ coincides with the Hausdorff dimension (up to a factor of 2) in dimension 1.
• In higher dimensions, it incorporates non-conformal geometry.
• It satisfies a Mañé–Manning type formula relating dimension, entropy, and Lyapunov exponents.
• It is characterized as the unique zero of a natural pressure function for expanding invariant measures.

The Specific Problem:
The paper investigates the variation of the volume dimension of the Julia set $J(f_t)$ for a family of holomorphic skew products $f_t$ near the monomial map $f_0(z, w) = (z^d, w^d)$. The family is defined as:
$$f_t(z, w) = \left(z^d, w^d + t \sum_{k=1}^d c_k(z) w^{d-k}\right)$$
where $c_k(z)$ are holomorphic functions. The goal is to derive an explicit second-order expansion of $VD_{f_t}(J(f_t))$ as $t \to 0$.

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2. Methodology

The proof follows a strategy adapted from McMullen's work but modified for the non-conformal, higher-dimensional setting. The methodology proceeds through the following steps:

A. Characterization via Pressure

The volume dimension $\delta_t$ is defined as the unique zero of the topological pressure function:
$$P(\delta_t \phi_t) = 0$$
where $\phi_t = -\log |\text{Jac } f_t|$ is the geometric potential. The problem reduces to computing the second derivative $\ddot{\delta}_0$ at $t=0$.

B. Thermodynamic Formalism and Variance

Using the implicit function theorem and properties of pressure, the authors relate $\ddot{\delta}_0$ to the variance of the infinitesimal deformation of the potential $\dot{\phi}_0$:
$$\ddot{\delta}_0 = \frac{\text{Var}(\dot{\phi}_0, m_0)}{2 \log d}$$
where $m_0$ is the measure of maximal entropy for the unperturbed map $f_0$.

C. Virtual Coboundaries and Böttcher Coordinates

A key insight, inspired by McMullen, is that $\dot{\phi}_0$ is not an arbitrary Hölder function but a virtual coboundary.

1. Conjugacy: The authors use Böttcher coordinates to define a conjugacy $H_t$ between the basins of infinity of $f_0$ and $f_t$.
2. Infinitesimal Deformation: They analyze the derivative of this conjugacy, denoted $\dot{v}$, which is a vertical vector field on the basin of infinity.
3. Relation: They establish that $\dot{\phi}_0 = -g + g \circ f_0$, where $g = \text{Re}(\partial_w v)$. This structure allows the variance of $\dot{\phi}_0$ to be computed via the asymptotic growth of $g$ near the boundary of the basin of infinity.

D. Asymptotic Energy Calculation

The variance is expressed as an "asymptotic energy" integral $I(\partial_w v)$ over circles $|w|=r$ as $r \to 1^+$:
$$\text{Var}(\dot{\phi}_0, m_0) = \frac{\log d}{2} \lim_{r \to 1^+} \frac{1}{|\log(r-1)|} \int_{S^1} \int_{|w|=r} \left| \frac{\partial v}{\partial w} \right|^2 d\text{Leb}_r(w) d\text{Leb}_1(z)$$

E. Explicit Computation

The authors solve the functional equation for $v$ explicitly in terms of the coefficients $c_k(z)$. By differentiating the conjugacy relation, they derive a series expansion for $\partial_w v$. Using the orthogonality of monomials in $L^2$ and the Birkhoff Ergodic Theorem, they compute the limit of the energy integral, yielding a sum involving the $L^2$ norms of the coefficients $c_k$.

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3. Key Contributions and Results

Main Theorem (Theorem 1.1):
The paper provides the explicit second-order expansion for the volume dimension of the Julia set of the skew product family $f_t$ as $t \to 0$:

$$VD_{f_t}(J(f_t)) = \frac{1}{2} + \frac{|t|^2}{16 d^2 \log d} \int_{S^1} \sum_{k=1}^d k^2 |c_k(z)|^2 \, d\text{Leb}_1(z) + O(|t|^3)$$

Key Technical Results:

1. Proposition 3.3: Establishes the formula linking the second derivative of the dimension to the variance of the potential deformation.
2. Proposition 3.5: Proves that the variance of a virtual coboundary equals a specific asymptotic energy integral of the generating function.
3. Proposition 3.7: Computes this energy integral explicitly for the skew product family, showing it depends on the weighted $L^2$ norms of the coefficient functions $c_k(z)$.

Structural Insights:

• The volume dimension function has a strict local minimum at the monomial map $(z, w) \mapsto (z^d, w^d)$, analogous to the 1D case.
• The "cost" of perturbation (the increase in dimension) is determined by the magnitude of the coefficients $c_k(z)$, weighted by $k^2$. Higher-degree terms in the $w$-expansion contribute more significantly to the dimension increase.

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4. Significance

1. Extension of Classical Theory: This work successfully generalizes the celebrated Ruelle-McMullen formulas from 1D conformal dynamics to 2D non-conformal holomorphic dynamics.
2. Validation of Volume Dimension: It demonstrates that the Volume Dimension is a robust dynamical invariant in higher dimensions, possessing smooth dependence on parameters and satisfying variational principles similar to the 1D Hausdorff dimension.
3. New Computational Tools: The paper introduces a method for computing dimension variations in non-conformal settings by utilizing Böttcher coordinates and virtual coboundaries, bypassing the need for conformal distortion theorems.
4. Foundation for Future Research: By establishing the analyticity and expansion of dimension in skew products, this work opens the door to studying more complex families of holomorphic endomorphisms in $\mathbb{C}^k$ and understanding the geometry of Julia sets in higher-dimensional parameter spaces.

In summary, Bianchi and He provide a rigorous analytical framework to quantify how the "size" (volume dimension) of chaotic sets in $\mathbb{C}^2$ changes under small perturbations, confirming that the monomial map represents a local minimum of complexity in this class of systems.