Computing multifractal spectra via simplicial measures

Abstract

Complex dynamical systems occur on many scales in the natural world, and serve as rich subjects of study. Examples include ecosystems, physiological systems, and financial markets. Simplified versions of these system can be described by dynamical systems. As such, understanding the qualitative behavior of dynamical systems provides an important window into real-world phenomena. In this manuscript we focus on the qualitative behavior described by the measure concentrated on the attractor of a dynamical system. A common way to study such complicated measures is through their multifractal spectra. We will describe a new method, developed to approximate the Sinai-Bowen-Ruelle measure on an attractor, that is based on the Vietoris-Rips complex. We use it to approximate various measures concentrated on a number of example sets, and demonstrate its efficacy by computing the corresponding multifractal spectra.