Triangular Constellations in Fractal Measures

TL;DR

This paper investigates the shapes of triangles formed by points sampling fractal measures generated by chaotic systems, revealing a phase transition in the probability distribution of a shape parameter related to the triangle's area.

Contribution

It introduces a novel analysis of triangle shape distributions in fractal measures and identifies a phase transition dependent on flow compressibility.

Findings

Probability density of the shape parameter is uniform below a critical flow compressibility.

Above the critical point, the distribution follows two distinct power laws.

The shape distribution exhibits a phase transition at a specific flow parameter.

Abstract

The local structure of a fractal set is described by its dimension $D$, which is the exponent of a power-law relating the mass ${\cal N}$ in a ball to its radius $\epsilon$: ${\cal N}\sim \epsilon^D$. It is desirable to characterise the {\em shapes} of constellations of points sampling a fractal measure, as well as their masses. The simplest example is the distribution of shapes of triangles formed by triplets of points, which we investigate for fractals generated by chaotic dynamical systems. The most significant parameter describing the triangle shape is the ratio $z$ of its area to the radius of gyration squared. We show that the probability density of $z$ has a phase transition: $P(z)$ is independent of $\epsilon$ and approximately uniform below a critical flow compressibility $\beta_{\rm c}$, but for $\beta>\beta_{\rm c}$ it is described by two power laws: $P(z)\sim z^{\alpha_1}$ when $1\gg z\gg z_{\rm c}(\epsilon)$, and $P(z)\sim z^{\alpha_2}$ when $z\ll z_{\rm c}(\epsilon)$.