Mostly Conjugate: Relating Dynamical Systems — Beyond Homeomorphism

A centerpiece of Dynamical Systems is comparison by an equivalence relationship called topological conjugacy. Current state of the field is that, generally, there is no easy way to determine if two systems are conjugate or to explicitly find the conjugacy between systems that are known to be equivalent. We present a new and highly generalizable method to produce conjugacy functions based on a functional fixed point iteration scheme. In specific cases, we prove that although the conjugacy function is strictly increasing, it is a.e. differentiable, with derivative 0 everywhere that it exists — a Lebesgue singular function. When applied to non-conjugate dynamical systems, we show that the fixed point iteration scheme still has a limit point, which is a function we now call a “commuter” — a non-homeomorphic change of coordinates translating between dissimilar systems. This translation is natural to the concepts of dynamical systems in that it matches the systems within the language of their orbit structures. We introduce methods to compare nonequivalent systems by quantifying how much the commuter functions fails to be a homeomorphism, an approach that gives more respect to the dynamics than the traditional comparisons based on normed linear spaces, such as L. Our discussion includes fundamental issues as a principled understanding of the degree to which a “toy model” might be representative of a more complicated system, an important concept to clarify since it is often used loosely throughout science.