On comparing dynamical systems by defective conjugacy: A symbolic dynamics interpretation of commuter functions

While the field of dynamical systems has been focused on properties which are invariant to “good” change of variables, namely conjugacy, which is an equivalence relationship, when using dynamical systems methods in science and modeling, there lacks a dynamical way to compare dynamical systems, even when they are in some sense “close.” In Skufca and Bolt (2007) [7] and Skufca and Bolt (2008) [8], we introduced mathematics to support a philosophy that two dynamical systems should be compared through a change of coordinates between them, that is, a commuter between them which may fail to be a homeomorphism. The progressive degree to which the commuter fails to be a homeomorphism defines what we call a homeomorphic defect. However, at the time of publication of these papers, there were limits in the mathematical technology requiring that the transformations be one-dimensional mappings and flows which are well described, for construction of the commuters by fixed point iteration, and further, difficulties in numerically computing defects in the more complicated one-dimensional cases, and further limits to higher-dimensional problems. Therefore, here we extend the theory to allow for multivariate transformations, with construction methods separate from the fixed point iteration, and new methods to compute defect. In the course of this work, we introduce several new technical innovations in order to cope with much more general problems. We introduce assignment mappings to understand and illustrate commuters in a broader setting. We discuss the role of symbolic dynamics and coding as related to commuters as well as defect measure. Further, we discuss refinement and convergence of a nested refinement of commuter representations. This work represents a step forward in the possibility of using the commuter and defects to judge model quality in those dynamical systems for which a symbolic dynamics, and hence a generating partition may be available; while finding a generating partition is a problem in its own right, we offer this work as further perspective for interpretation of the meaning of commuters and defect measure.