Mathematical Sciences: An Experimental Tool for Topological Surface Dynamics

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9626884 Menasco The investigators develop a computer program for studying topological properties of maps of punctured surfaces. A central component of the software tool is the Bestvina-Handel algorithm for the Thurston-Nielsen classification of surface homoeomorphisms, which yields not only the classification, but also a canonical representative - a "train-track map" - of the isotopy class to which the homeomorphism belongs. Experimentation in this field, essential to the formulation of theorems and conjectures, is currently hampered by the very large amount of labor entailed in classifying and analyzing even a single example of a surface homeomorphism. The software removes this roadblock. It allows the researcher to rapidly generate train-track maps, and representations and characterizations (both algebraic and graphical) of their suspensions. The software is made available via the World Wide Web. One studies dynamical systems to explain and make predictions about a vast range of phenomena, from the motions of the stars and planets, to the circulation of the Earth's atmosphere, to the behavior of machinery such as drilling rigs and human hearts, to the self-propulsion of bacteria. In describing such phenomena, which are repetitive in character, models in the form of iterated maps from some space to itself frequently have great explanatory and predictive power. Perhaps the simplest class of realistic models is that of continuous maps of two-dimensional surfaces. Even simple members of this class can exhibit a bewildering complexity of behavior, and many questions about these systems have yet to be answered. Maps of punctured surfaces are of interest because of their bearing on the relationships among periodic orbits and on long-standing problems of the theory of knots. The purpose of this project is to develop a software tool for experimentation in the topological properties of maps of punctured surfaces. Topological properties, those that ar e persistent under distortions of the surface or map, are the most fundamental. The software is made available via the World Wide Web.