Volume 107, Number 5, September 2014
|Number of page(s)
|05 September 2014
Triangular constellations in fractal measures
Department of Mathematics and Statistics, The Open University - Walton Hall, Milton Keynes, MK7 6AA, England, UK
Received: 22 May 2014
Accepted: 18 August 2014
The local structure of a fractal set is described by its dimension D, which is the exponent of a power-law relating the mass in a ball to its radius . It is desirable to characterise the 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 ε and approximately uniform below a critical flow compressibility , which we estimate. For the distribution appears to be described by two power laws: when , and when .
PACS: 05.45.Df – Fractals / 47.53.+n – Fractals in fluid dynamics / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion
© EPLA, 2014
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.