Flexible Hausdorff Dimension in Jordan Curves

• Flexible curves are Jordan curves with tunable Hausdorff dimensions between 1 and 2, constructed via log‐singular weldings and quasiconformal iterations.
• The construction employs iterative schemes with capacity‐thin sets and s‐additive square selections to precisely control geometric properties and area measures.
• These curves exemplify non‐injectivity in conformal welding and serve as test cases for studying removability, dimension distortion, and fractal phenomena in complex analysis.

A flexible curve of Hausdorff dimension $s$ is a Jordan curve in the complex sphere $\widehat{\mathbb{C}}$ whose geometric properties can be finely tuned with respect to both conformal welding and fractal dimension. The recent development in the theory provides a systematic framework for prescribing the Hausdorff dimension $s\in[1,2]$ of such curves, while also controlling their conformal welding correspondence. These constructions fundamentally engage the quasiconformal geometry of the plane, properties of logarithmic capacity, and the structure of circle homeomorphisms known as log-singular weldings. Flexible curves, which form a residual subset in the space of all Jordan curves, serve as a focal point in understanding the non-injectivity of the welding correspondence, the phenomenon of positive-area Jordan curves, and conformal removability (Rodriguez, 20 Jan 2026).

1. Foundational Concepts: Jordan Curves, Welding, and Hausdorff Dimension

A Jordan curve $\gamma\subset\widehat{\mathbb{C}}$ partitions the sphere into complementary components $\Omega$ (bounded) and $\Omega^*$. By the Riemann mapping theorem, there exist conformal maps $f: \mathbb{D} \to \Omega$ and $g: \mathbb{D}^* \to \Omega^*$, normalized so that $f(0)=a\in\Omega$, $g(\infty)=\infty$, and both extend continuously to the boundary $\mathbb{S} = \partial\mathbb{D}$. The conformal welding $h = g^{-1}\circ f: \mathbb{S} \to \mathbb{S}$ is an orientation-preserving homeomorphism, defined up to Möbius automorphisms. The welding correspondence $\mathcal{W}: [\gamma] \to [h]$ associates Möbius classes of curves to those of weldings.

Hausdorff dimension is defined via coverings by disks of prescribed radii $r_j$, with the $s$-Hausdorff measure $\mathcal{H}_s(E)$ capturing the scaling law: $$\mathcal{H}_s(E) = \lim_{\delta\to 0} \inf \left\{ \sum_j r_j^s : E\subset \bigcup_j D(z_j, r_j),\ r_j\leq\delta \right\}$$ and $\dim_H(E)$ is the infimum $s$ at which $\mathcal{H}_s(E)=0$. For a curve $\Gamma$, the notation $d_H(\Gamma)$ is used for its Hausdorff dimension (Rodriguez, 20 Jan 2026).

2. Flexible Curves and Log-Singular Weldings

A Jordan curve $\gamma\subset\mathbb{C}$ is called flexible (Bishop's definition) if for every Jordan curve $\tilde{\gamma}$ and every $\epsilon>0$, there exists a homeomorphism $\Phi:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$, conformal off $\gamma$, so that the Hausdorff distance $d_H(\Phi(\gamma),\tilde{\gamma})<\epsilon$, and the images of two points from each complementary component can be prescribed.

Equivalently, a curve is flexible if its welding $h$ is log-singular: there exists a Borel set $E\subset\mathbb{S}$ of logarithmic capacity zero such that $h(\mathbb{S}\setminus E)$ also has logarithmic capacity zero. This property establishes a deep link between fine potential-theoretic null sets and topological flexibility (Rodriguez, 20 Jan 2026).

3. Existence Theorem and Dimensional Control

Main Existence Theorem

Let $h:\mathbb{S}\to\mathbb{S}$ be a log-singular circle homeomorphism, and let $s\in[1,2]$. Then there exists a flexible curve $\Gamma\subset\widehat{\mathbb{C}}$ whose conformal welding is $h$ and whose Hausdorff dimension satisfies $d_H(\Gamma)=s$ (Rodriguez, 20 Jan 2026).

Positive-Area Companion

Under the same hypothesis on $h$, there exists a Jordan curve $\tilde{\Gamma}$ with conformal welding $h$, such that $\tilde{\Gamma}$ has positive planar Lebesgue area (which is equivalent to $\dim_H(\tilde{\Gamma})=2$).

This result not only enables construction of flexible curves of any prescribed dimension in $[1,2]$, but also shows that, for a given log-singular welding, one can realize this welding by both a zero-area and a positive-area Jordan curve.

4. Construction Scheme and Technical Overview

The construction of a flexible curve with prescribed dimension proceeds via an iterative quasiconformal scheme:

• Begin with initial conformal maps $(f_0,g_0)$. Iteratively, produce sequences $(f_n,g_n)$ of quasiconformal maps with maximal dilatations $K_n\uparrow K<\infty$ and the welding $h$; the gap $$\delta_n=\sup_{\xi\in\mathbb{S}}|f_n(\xi)-g_n(h(\xi))|$$ decays geometrically.
• At each stage, select a capacity-thin set $E_n\subset\mathbb{S}$ and corresponding “star-shaped” conformal region $W_n$ with slits encoding these arclengths.
• Quadilaterals $Q_{n,k}$ and conformal embeddings $E_{n,k}$ are employed to control the geometry locally. Small-dilatation qc maps $\alpha_{n,k}$ correct mismatches, and new Beltrami coefficients $\mu_{n+1}$ are supported in thinner and thinner strips, maintaining $\|\mu_{n+1}\|_\infty\ll 1$.
• Flexibility is enforced by steering $E_n$ to “miss” arcs relevant to a target curve. The construction ensures that, for any target Jordan curve, the limiting flexible curve can be mapped near it in the Hausdorff metric by a homeomorphism conformal off the original curve.
• Hausdorff dimension $s$ is controlled via two mechanisms:
  • For $s=2$ (positive-area), choose embeddings so that each $Q_{n,k}$ is missing a small proportion $a_n$ of its area, with $\prod a_n>0$, ensuring $m(\Gamma)>0$.
  • For $1$n(s)$ “$s$-additive” squares of side $x(s)$ in each $Q_{n,k}$ so that $\sum_{j=1}^{n(s)} (\sqrt{2}x(s))^s=1$, with separation properties guaranteeing dimensional regularity. Covering the remainder by disks of radii $r_{n,j}$ with $\sum_j r_{n,j}^s\leq 2^{-n}$ supports a Frostman-type argument showing $$0<\mathcal{H}_s(\Gamma)<\infty\implies \dim_H(\Gamma)=s$$.

The entire process is stabilized using distortion inequalities: Astala’s theorem gives

$$\frac{\dim_H(E)}{1+C\|\mu\|_\infty} \leq \dim_H(f(E)) \leq (1+C\|\mu\|_\infty)\dim_H(E)$$

and because $\|\mu_n\|_\infty\to 0$ away from vanishing neighborhoods of $\Gamma$, the final Jordan curve after the global straightening map $H$ has precisely the prescribed Hausdorff dimension (Rodriguez, 20 Jan 2026).

5. Welding Non-injectivity, Removability, and Residual Flexibility

The existence of distinct flexible curves (of different dimension, or with positive area) having the same conformal welding demonstrates non-injectivity in the welding correspondence for a residual set of Jordan curves. In particular, given a flexible curve $\Gamma$, there is a global homeomorphism $\varphi:\mathbb{C}\to\mathbb{C}$, conformal off $\Gamma$, such that $\varphi(\Gamma)$ has positive area. This answers a special case of the conjecture that non-conformally removable sets admit such deformations. For flexible curves, which are residual by Pugh–Wu, this establishes that “most” Jordan curves admit uncountably many non-Möbius-equivalent representatives with the same welding but varying Hausdorff dimension in $[1,2]$ (Rodriguez, 20 Jan 2026).

A Jordan curve $\gamma$ is conformally removable if every homeomorphism of $\widehat{\mathbb{C}}$ conformal off $\gamma$ is Möbius. Bishop’s flexible curves are non-removable. The results confirm the conjectured equivalence between non-removability and non-injectivity of the welding correspondence for a generic class.

6. Parameter Selection, Representative Examples, and Related Models

While no explicit formula for a flexible curve of given dimension is provided, the construction prescribes parameter choices at each iteration:

• The capacity set $E_n$ is made small so that slit-maps $\varphi_n$ image $\mathbb{S}$ into regions of radius $\exp(A_n/N_n)$, $A_n\to\infty$.
• For intermediate $s$, the $n(s)$, $x(s)$ are chosen to satisfy $4 n(s) (\sqrt{2}x)^s =1$, with $x\ll 1$.
• The separation parameter $P(s) = \dfrac{25}{4\cdot 2^{s/2} x^{2-s}}$ governs the lattice occupation and ensures uniform separation at scale $x(s)$.

This systematic, parameter-driven construction yields the full spectrum $s\in(1,2)$.

By comparison, the study of Fibonacci word fractal curves demonstrates that self-similar limit sets arising from combinatorially constructed polygonal curves (with prescribed rule depending on a drawing angle $\alpha\in[0,\pi/2]$) also exhibit a Hausdorff dimension $s(\alpha)$ computable by the formula $s(\alpha) = \dfrac{\ln(\sqrt{5}-2)}{\ln R(\alpha)}$, with $R(\alpha)$ an explicit function of $\cos\alpha$. Each of these fractal curves interpolates between a line segment ($s=1$) and the classical Fibonacci “U–curve” ($s\approx 1.637$ for $\alpha=\pi/2$) (Hoffman et al., 2016).

7. Broader Implications, Applications, and Outlook

The existence and control of flexible curves of arbitrary Hausdorff dimension substantiate the richness of the space of Jordan curves in planar quasiconformal geometry. These results provide:

• The first systematic constructions of non-injectivity in welding outside trivial positive-area scenarios.
• Resolution (in the residual flexible case) of the conjecture connecting non-removability with the non-injectivity of conformal welding.
• Flexible curves as universal sources for dense approximation in the Hausdorff metric and as test objects for removability and dimension-distortion problems.

Techniques such as quasiconformal iteration, the use of capacity-thin sets, and $s$-additive combinatorial models may be adaptable to further studies in harmonic measure, geometric function theory, and the fine geometry of fractal sets.

The interplay between geometrically flexible Jordan curves and log-singular weldings continues to motivate developments connecting fractal dimension, conformal structure, and the theory of removability and rigidity in complex analysis (Rodriguez, 20 Jan 2026).