A Heisenberg Subdivision Scheme with Central Smoothness Loss

Technical Summary: A Heisenberg Subdivision Scheme with Central Smoothness Loss

Problem Statement
Subdivision schemes are fundamental tools for generating smooth curves and surfaces from discrete control data. While linear subdivision in Euclidean space is well-understood, many applications involve data taking values in non-Euclidean manifolds or Lie groups, where component-wise linear refinement is geometrically inappropriate. Existing approaches, such as those based on proximity frameworks (Wallner and Dyn) or intrinsic log-exp constructions, generally aim to prove that a nonlinear scheme inherits the smoothness of its linear reference under specific quantitative closeness assumptions.

This paper addresses a gap in the theory of noncommutative subdivision. Specifically, it investigates whether a geometrically natural, explicit interpolatory rule defined directly on the three-dimensional Heisenberg group ($Heis_3$)—the simplest noncommutative model geometry—preserves the smoothness of its Euclidean counterpart. The authors construct a scheme where the central coordinate is corrected by a term derived directly from the group law and analyze the regularity of the resulting limit curve.

Methodology
The authors define an explicit interpolatory subdivision operator, $S_H$, acting on control polygons in $Heis_3$. The group law is given by:
$$(x, y, z) \cdot (x', y', z') = \left(x + x', y + y', z + z' + \frac{1}{2}(xy' - yx')\right)$$

The refinement rule is constructed as follows:

1. Horizontal Coordinates ($x, y$): Refined using the classical, linear four-point scheme of Dyn, Gregory, and Levin ($L_4$) with parameter $\omega = 1/16$.
2. Central Coordinate ($z$): Refined by applying the linear four-point mask to the existing $z$-values and adding a correction term $R^k_i$. This correction is derived explicitly from the group law to account for the non-commutativity when inserting a new point between two existing points.
   $$R^k_i = \frac{1}{2}(x^k_i b^k_i - y^k_i a^k_i)$$
   where $a^k_i$ and $b^k_i$ are the horizontal displacements generated by the four-point rule.

The analysis relies on the block-triangular structure of the refinement map, which decouples the horizontal evolution from the central one. The core of the analysis involves deriving an exact two-scale recursion for the scaled central first differences ($e^k_i = 2^k(z^k_{i+1} - z^k_i)$). This recursion takes the form:
$$e^{k+1} = S e^k + \hat{g}^{k+1}$$
where $S = 2D_1$ is the scaled first-difference operator (which is not a contraction) and $\hat{g}^{k+1}$ is a forcing term derived from the scaled correction $F^k_i = 2^k R^k_i$.

Key Contributions and Results

1. Explicit Noncommutative Scheme: The paper provides a fully explicit, coordinate-based interpolatory subdivision scheme for $Heis_3$. Unlike intrinsic log-exp schemes, this rule is not left-equivariant but allows for direct geometric interpretation and closed-form analysis of the correction term.

2. Decoupled Regularity:

   • Horizontal Limit: The horizontal components $(X, Y)$ converge to the classical $C^1$ four-point limit, independent of the central data.
   • Central Limit: The central component $Z$ converges uniformly to a continuous function. However, it exhibits a distinct loss of regularity compared to the Euclidean case.

3. Zygmund Regularity and Loss of $C^1$:

   • The authors prove that the central limit $Z$ belongs to the Zygmund class $\Lambda^*$, possessing a modulus of continuity of order $O(h \log(1/h))$.
   • Crucially, the scaled first differences of the central coordinate grow linearly with the refinement level $k$ (i.e., $\sup_i 2^k |z^k_{i+1} - z^k_i| = O(k)$).
   • Under an explicit, verifiable signed non-cancellation condition on the detail coefficients (which holds for typical non-degenerate data), the authors prove a matching lower bound $\Theta(k)$. This growth implies that the limit function is not continuously differentiable ($Z \notin C^1$).

4. Mechanism of Smoothness Loss:
   The paper identifies the precise mechanism for this regularity loss. The forcing term in the central difference recursion is the undifferenced scaled correction. Because the limit of the scaled correction $2^k R^k_i$ converges to a generally non-zero function $f(t) = \frac{1}{4}(XY' - YX')$, the forcing amplitude does not decay across scales. This persistent forcing is fed through the operator $S$, which has a neutral component (it preserves alternating sequences). Consequently, a fresh oscillation of fixed amplitude is injected at every refinement level, causing the first differences to accumulate linearly.

5. Numerical Verification:
   Numerical experiments confirm the theoretical findings. The central coordinate exhibits a visibly rough, non-smooth profile. The scaled first differences grow linearly with the refinement level, while the second-difference quotient remains bounded, confirming the Zygmund regularity and the failure of $C^1$ smoothness.

Significance and Claims
The paper positions this result as a negative regularity example within nonlinear subdivision theory. It does not contradict existing positive smoothness-transfer results (such as those based on proximity or log-exp frameworks) but rather operates outside their hypotheses.

• Failure of Proximity: The authors demonstrate that the Heisenberg correction satisfies only a linear bound relative to the horizontal increments, whereas proximity-based $C^1$ transfer requires a quadratic bound ($O(\|\Delta u\|^2)$). The failure of this quadratic condition is not a technical artifact but a faithful reflection of the genuine non-smoothness of the limit.
• Cautionary Lesson: The work serves as a warning for coordinate-based or basepointed group-law constructions. A correction that is geometrically natural and harmless at a single refinement step can destroy the regularity of the limit curve if it is injected with non-decaying amplitude at every scale.
• Scope: The result is specific to this explicit, non-equivariant coordinate scheme. It does not claim that all Lie-group subdivision schemes fail to preserve smoothness, but it highlights that the specific mechanism of using the group law to generate a first-order correction in a noncommutative setting can lead to a loss of one logarithmic order of smoothness (from $C^1$ to Zygmund).

In summary, the paper establishes that while the horizontal geometry remains smooth, the noncommutative correction in the central channel introduces a persistent forcing that prevents the limit curve from being $C^1$, resulting instead in a continuous, Zygmund-regular limit.