Regularity of Commuter Functions for Homeomorphic Defect Measure in Dynamical Systems Model Comparison

In the field of dynamical system, conjugacy describes an equivalent relation between two dynamical systems. In our work, we are dealing with mostly conjugacy, which relates two dynamical systems that are not necessarily conjugate. We generate a function called ”commuter” based on a fixed point iteration scheme. The resulting ”commuter” is a nonhomeomorphic change of coordinates translating between two systems. And we can determine the amount of failing to be conjugacy, which we call homeomorphic defect, by studying the properties of commuters. We consider the function space Lp[0, 1], with 1 ≤ p < ∞, and the norm is given by the standard Lp norm. We derive a contractive operator which will give a limit point from the commuting relationship even when applied to nonconjugate systems. We discuss the measurability of commuters. Specially, when studying behaviors of commuters between full symmetric tent map and short symmetric tent map, we show that the commuter is monotonely convergent to identity function as the height of the short one is going to 1. At last, we also give a computation error analysis for our computation method in producing commuters.