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On the Steiner tree connecting a fractal set

Emanuele Paolini, Eugene Stepanov

TL;DR

This work addresses the Steiner problem for a planar self-similar fractal boundary $C$ with positive Hausdorff dimension, seeking the shortest planar network $S$ such that $S\cup C$ is connected. It constructs an explicit infinite self-similar binary tree $Σ$ via the IFS $f_±(z)=1+\lambda e^{± i\pi/3} z$, with leaves $A$ and trunk $[O,T]$, and defines $Σ=[O,T]\cup f_+(Σ)\cup f_-(Σ)$; the leaves are $A$ and $C=\{O\}\cup A$, making $Σ$ an irreducible Steiner solution. A universal tree $Σ_u=Σ\cup φ(Σ)\cup φ^2(Σ)$ is also defined using a $120^\circ$ rotation, and the paper proves that, for sufficiently small $\lambda$, $Σ$ minimizes the Steiner length for $A$ (and $Y\cup A$) and $Σ_u$ minimizes for $A_u$, while also establishing a universality property: every abstract binary tree can be realized as a Steiner subtree of $Σ_u$. The results provide explicit, scalable optimal networks for fractal data and offer a symmetry-based framework for understanding Steiner trees in infinite settings, leveraging calibrations and self-similarity to obtain minimality and uniqueness. Overall, the paper advances explicit constructions and universality principles in planar Steiner problems with fractal data, connecting finite-and-infinite data regimes through a common symmetry-driven methodology.

Abstract

We construct an example of an infinite planar embedded self-similar binary tree $Σ$ which is the essentially unique solution to the Steiner problem of finding the shortest connection of a given planar self-similar fractal set $C$ of positive Hausdorff dimension. The set $C$ can be considered the set of leaves, or the ``boundary``, of the tree $Σ$, so that $Σ$ is an irreducible solution to the Steiner problem with datum $C$ (i.e. $Σ\setminus C$ is connected).

On the Steiner tree connecting a fractal set

TL;DR

This work addresses the Steiner problem for a planar self-similar fractal boundary with positive Hausdorff dimension, seeking the shortest planar network such that is connected. It constructs an explicit infinite self-similar binary tree via the IFS , with leaves and trunk , and defines ; the leaves are and , making an irreducible Steiner solution. A universal tree is also defined using a rotation, and the paper proves that, for sufficiently small , minimizes the Steiner length for (and ) and minimizes for , while also establishing a universality property: every abstract binary tree can be realized as a Steiner subtree of . The results provide explicit, scalable optimal networks for fractal data and offer a symmetry-based framework for understanding Steiner trees in infinite settings, leveraging calibrations and self-similarity to obtain minimality and uniqueness. Overall, the paper advances explicit constructions and universality principles in planar Steiner problems with fractal data, connecting finite-and-infinite data regimes through a common symmetry-driven methodology.

Abstract

We construct an example of an infinite planar embedded self-similar binary tree which is the essentially unique solution to the Steiner problem of finding the shortest connection of a given planar self-similar fractal set of positive Hausdorff dimension. The set can be considered the set of leaves, or the ``boundary``, of the tree , so that is an irreducible solution to the Steiner problem with datum (i.e. is connected).
Paper Structure (4 sections, 16 theorems, 30 equations, 3 figures)

This paper contains 4 sections, 16 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Preparatory results
  4. Main results

Key Result

Proposition 2.1

Let $C\subset \mathbb R^n$ be a compact set. Then $\mathcal{M}(C)\neq \emptyset$. If $S\in \mathcal{M}(C)$ and $\mathcal{H}^1(S)<+\infty$ then the following statements hold:

Figures (3)

  • Figure 1: The self-similar tree with a countable number of triple junctions and an uncountable number of leaves which are the limit points of the triple junctions. On the left the universal tree $\Sigma_u$, on the right a single branch $\Sigma$, with a fixed root in the origin.
  • Figure 2: The situation in Lemma \ref{['lm:angle']}.
  • Figure 3: Constructions and notation in the proof of Lemma \ref{['lm:branching']} and Theorem \ref{['th:main']}. The triangle $\Delta$ from the proof of Lemma \ref{['lm:branching']} is highlighted.

Theorems & Definitions (33)

  • Proposition 2.1: known facts about minimizers
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 23 more