Fractal Dimension for Measures via Persistent Homology
Fractal dimensions give a way to describe objects that display multiscale complexity in their structure. Fractal objects appear in a wide variety of contexts from chaotic dynamical systems to distributions of earthquakes. While fractal dimensions are most classically defined for a space, there are a variety of fractal dimensions for measures, like the Hausdorff dimension. In this talk, I will define a fractal dimension based on persistent homology. This fractal dimension can be estimated computationally and works for arbitrary probability measures on metric spaces. We will look at examples and will discuss several interesting related conjectures.