Filtrations Indexed by Attracting Levels and their Applications

Abstract

We introduce a new class of filtrations indexed by attracting levels in dynamical systems, providing novel inputs for persistent homology and related methods in topological data analysis. These filtrations quantify, in a forward direction, the sensitivity of trajectories with respect to attractors under perturbations and, in a backward direction, the perturbation magnitude at which attraction breaks down. The construction applies not only to maps on metric spaces but also to general partial maps with cost functions, yielding a filtration-theoretic framework with connections to algebraic topology. This generality ensures complementary filtrations when terminal states are good or bad, inducing natural decompositions of the underlying space. As an illustration, we apply the framework to ensemble forecasts of tropical cyclones, where the filtrations identify regions of heightened sensitivity, demonstrating the potential of our approach as a new tool for topological data analysis applied to dynamical systems.