Reeb Spaces and the Robustness of Preimages

Abstract

We study how the preimages of a mapping f : X &rarr Y between manifolds vary under perturbations. First, we consider the preimage of a single point and track the history of its connected component as this point varies in Y. This information is compactly represented in a structure that is the generalization of the Reeb graph we call the Reeb space. We study its local and global properties and provide an algorithm for its construction. Using homology, we then consider higher dimensional connectivity of the preimage. We develop a theory quantifying the stability of each homology class under perturbations of the mapping f . This number called robustness is given to each homology class in the preimage. The robustness of a class is the magnitude of the perturbation necessary to remove it from the preimage. The generality of this theory allows for many applications. We apply this theory to quantify the stability of contours, fixed points, periodic orbits, and more.