A Computational Approach to Measuring Homeomorphic Defect

A Computational Approach to Measuring Homeomorphic Defect by Scott M. LaLonde Master of Science in Mathematics Clarkson University An important concept in the field of dynamical systems is the notion of conjugacy. Two dynamical systems are said to be conjugate if their dynamics are topologically equivalent. In other words, there is a homeomorphism between the underlying spaces which preserves the dynamics of the two systems. In this thesis we will be discussing an extension of this idea called mostly conjugacy. In the context of mostly conjugacy, we deal with functions called commuters. These relate two dynamical systems that are not necessarily conjugate. We can determine the amount by which two dynamical systems fail to be conjugate by studying certain properties of their associated commuter. As commuters are not generally homeomorphisms, we will do so by studying a quantity called the homeomorphic defect. The work presented here is devoted largely to developing computational techniques for measuring this defect. In particular, we will construct an algorithm for approximating the Lebesgue measure of subsets of Rn which will rely heavily on the concept of Monte Carlo integration. Once we have constructed the algorithm, we will present results from its deployment on various benchmark sets. We will analyze these results and use them to present arguments for the validity of the algorithm. Finally, we will discuss open questions and possible future work that can be done to achieve the goal of measuring homeomorphic defect to a reasonable degree of accuracy.