Lacunarity Transition

TL;DR

This paper introduces a simplified model to understand the formation of large empty regions in the phase space of chaotic systems, revealing a lacunarity transition linked to the statistics of Wiener process maxima.

Contribution

It extends the skinny bakers' map to model sparsely occupied phase space regions and connects void size distribution to Wiener process statistics, demonstrating a lacunarity transition.

Findings

Identification of a lacunarity transition in the model

Mapping void size distribution to Wiener process maxima

Observation of persistent empty regions with increasing trajectories

Abstract

Experiments investigating particles floating on a randomly stirred fluid show regions of very low density, which are not well understood. We introduce a simplified model for understanding sparsely occupied regions of the phase space of non-autonomous, chaotic dynamical systems, based upon an extension of the skinny bakers' map. We show how the distribution of the sizes of voids in the phase space can be mapped to the statistics of the running maximum of a Wiener process. We find that the model exhibits a lacunarity transition, which is characterised by regions of the phase space remaining empty as the number of trajectories is increased.