Directional Dilatation in Geometry & Analysis

• Directional dilatation is a measure that quantifies the stretching or contraction of mappings along specified directions, underpinning studies in flat geometry, quasiconformal analysis, and computational shape optimization.
• It plays a critical role in dilation surfaces where holonomy and saddle connections govern contraction rates and induce Morse–Smale dynamical behavior.
• Analytical formulations using normal directional dilatation sharpen modulus estimates and support efficient boundary-face dilation methods for PDE-constrained design problems.

Directional dilatation, in its technical usage across geometry, analysis, and computational mathematics, quantifies how a mapping or geometric structure stretches or contracts in specific directions. The phenomenon arises prominently in the study of dilation surfaces, shape optimization via unfitted finite element discretizations, and higher-dimensional modulus estimates for quasiconformal mappings. Directional dilatation is fundamentally linked to the linear holonomy of closed geodesics, return maps of foliations, the local structure of differential mappings, and the shape derivative calculus for boundary perturbations.

1. Dilation Surfaces and Directional Dilatation

A dilation surface is a compact topological surface $X$ (possibly with boundary) and a finite set of marked points $\Sigma$ (singularities), together equipped with a dilation atlas. Local charts $(U_i, \phi_i)$ cover $X \setminus \Sigma$ with transition functions of the form

$$z \longmapsto a_{ij} z + b_{ij}, \qquad a_{ij} \in \mathbb{R}_+^*,\ b_{ij} \in \mathbb{C}$$

At singularities $p \in \Sigma$, the local geometry is a flat cone with total angle $2\pi i_p$ and identification via $z \mapsto \rho_p z$, $\rho_p > 0$ (Tahar, 2021, Ghazouani, 2019).

The structure generalizes translation surfaces (where all $a_{ij} = 1$). In strict dilation surfaces (with some $a_{ij} \neq 1$), the geometry admits nontrivial directional dilatation phenomena.

Directional dilatation in this context is the linear part $\lambda$ of the holonomy of a closed geodesic $\gamma$; for $\gamma$ oriented so that $0 < \lambda < 1$, leaves in the prescribed direction $\theta(\gamma)$ contract at rate $\lambda$ (or expand when traversed "backwards"). This exponential contraction governs the asymptotic dynamics of directional flows on dilation surfaces (Tahar, 2021).

2. Topological and Dynamical Manifestations

Saddle connections (straight segments between singularities) play a crucial role. A horizon saddle connection $L$ is characterized by the existence of an integer $k$ such that no trajectory intersects $L$ more than $k$ times; this creates a "horizon" beyond which trajectories cannot repeatedly cross, a phenomenon absent in translation surfaces (Tahar, 2021). The presence of a horizon saddle connection implies a fundamentally different dynamical structure: in dilation surfaces, the directions of hyperbolic closed geodesics (those with linear holonomy $\lambda \neq 1$) form a dense subset of $\mathbb{S}^1$, as formalized in:

$$\overline{\{\theta \in \mathbb{S}^1\! :\! \exists\;\gamma \subset X\ \text{hyperbolic closed geodesic with}\ \theta(\gamma) = \theta \}} = \mathbb{S}^1$$

The equivalence between the density of hyperbolic directions and Morse–Smale dynamics is rigorously established: on strict dilation surfaces, an open dense set of directions exhibits Morse–Smale behavior—every trajectory either terminates at a singularity or converges to an attracting/repelling periodic orbit associated to directional dilatation (Tahar, 2021, Ghazouani, 2019).

3. Analytical Formalism: Modulus Estimates and Radial Dilatation

In the analysis of quasiconformal mappings, especially for the detection of cavitation in higher dimensions, directional dilatation is developed as a pair of distinct invariants: angular dilatation $D_f$ and normal (radial) directional dilatation $Q_f$ (Golberg et al., 21 Dec 2025).

Given $f: 0 < |x| < 1 \rightarrow \mathbb{R}^n$, at regular points $x$ (with respect to a pole $x_0$), define: $$u = \frac{x - x_0}{|x - x_0|},\quad \partial_h f(x) = Df(x) h\ (h \in S^{n-1})$$ The classical angular dilatation (with respect to $x_0$) is

$$D_f(x, x_0) = \frac{J_f(x)}{\ell_f(x, x_0)^n},\quad \ell_f(x, x_0) = \min_{h \in S^{n-1}} \frac{|\partial_h f(x)|}{|h \cdot u|}$$

The novel normal (radial) directional dilatation is

$$Q_f(x, x_0) = \left( \frac{|\partial_u f(x)|^n}{J_f(x)} \right)^{1/(n-1)}$$

These invariants admit the chain of inequalities

$$\frac{1}{K_f(x)} \leq \cdots \leq Q_f(x, x_0) \leq \cdots \leq L_f(x)$$

where $K_f$ and $L_f$ encapsulate classical quasiconformal distortion.

The precise two-sided modulus bounds for curve families in annuli (Theorem 2.12) are: $$\left(\int_{\mathcal{A}(r, R)} p\bigl(\tfrac{x}{|x|}\bigr)^n Q_f(x)\frac{dm_n(x)}{|x|^n}\right)^{1-n} \leq M\bigl(f(\Gamma)\bigr) \leq \int_{\mathcal{A}(r, R)} \rho\bigl(|x|\bigr)^n D_f(x)\,dm_n(x)$$ where $p$ and $\rho$ are non-negative measurable test functions. The normal dilatation $Q_f$ sharpens the lower bounds, enabling the detection of cavitation phenomena previously inaccessible using only angular estimates (Golberg et al., 21 Dec 2025).

4. Computational Shape Calculus: Boundary Dilatation Approach

Directional dilatation also manifests in shape optimization, specifically in shape calculus for PDE-constrained design problems. The boundary-face dilation approach, based on level-set perturbations, employs directional dilatation localized to boundary neighborhoods (Berggren, 2022). For a domain $\Omega = \{ x \in D \mid \phi(x) < 0 \}$ with boundary $\partial\Omega = \{ \phi = 0 \}$, a vector field $V$ and cutoff $\chi$ induce the perturbed level-set

$$\phi_\epsilon(x) = \phi(x) + \epsilon \chi(x) V(x) \cdot n_\phi(x)$$

where $n_\phi$ is the outward unit normal. The first-order shape derivative of a volume integral $J(\Omega)$ under boundary-face dilation yields the boundary-supported formula: $$\left.\frac{d}{d\epsilon} J(\Omega_\epsilon)\right|_{\epsilon = 0^+} = -\int_{\partial\Omega} j(x) \chi(x) (V \cdot n_\phi)\, ds$$ This method circumvents the need for mesh deformation (required in classical domain transformation approaches), works with minimal regularity assumptions, and is naturally compatible with unfitted finite element discretizations (Berggren, 2022). Directional dilatation here refers to the normal displacement induced by $V$ at the boundary.

5. Geometric and Dynamical Consequences

Directional dilatation in dilation surfaces is responsible for the emergence of non-minimal, non-uniquely ergodic dynamics. Morse–Smale behavior—where generic directional flows exhibit attracting or repelling limit cycles—is typical in strict dilation surfaces, constituting the generic scenario by openness and density in moduli space (Tahar, 2021, Ghazouani, 2019). In genus-two surfaces built from two chambers joined by a horizon saddle connection, directional dilatation is immediately visualized in the contraction factor of the cylinder's holonomy.

In higher-dimensional quasiconformal analysis, normal dilatation $Q_f$ enables sharp modulus estimates capable of distinguishing topological cavitation; for example, in mapping a punctured ball onto a ring, the positivity of the $I_Q(f)$ integral, involving $Q_f$, guarantees cavitation at the origin (Golberg et al., 21 Dec 2025). Absence of cavitation is characterized by the unboundedness of $Q_f$ along rays.

Boundary-face dilation for shape derivatives provides an efficient, non-intrusive methodology for gradient-based optimization, handling discontinuous integrand regularity and domain boundaries directly.

6. Connections Across Disciplines

Directional dilatation unifies several distinct mathematical themes:

• In flat surface geometry, it encodes holonomy-based contraction/expansion and underpins the structure of moduli spaces, dynamical classes, and geometric flows such as the Teichmüller deformation. Renormalization via diagonal $SL_2(\mathbb{R})$ action reveals scaling of directional dilatation in coordinate charts (Ghazouani, 2019).
• In modulus estimates for elasticity and geometric analysis, it supplies refined bounds sensitive to stretching in specific directions, thereby detecting topological features of mappings inaccessible by classical invariants alone (Golberg et al., 21 Dec 2025).
• In computational shape optimization, boundary-localized directional dilatation yields efficient, robust design formulas compatible with unfitted finite element approaches—foundational for contemporary numerical PDE solutions in varied domains (Berggren, 2022).

This suggests directional dilatation is increasingly recognized as a fundamental invariant transcending its original context and facilitating new methodologies in analysis, geometry, and scientific computing.