Mathematical Sciences: Complex Dynamical Systems and Mobius Groups

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9400999 Hinkkanen This award supports mathematical research on problems arising in the study of complex dynamical systems, Mobius groups and their topological counterparts. The work focuses on complex functions of a complex variables and the domains on which these functions are defined. The first set of problems concerns sets of rational functions and the semigroups they determine via composition. These semigroups are natural extensions dynamical systems derived from iterations of a single function. The theory leads to a great many dynamical, combinatorial and algebraic questions and exhibits a rich structure involving properties and dynamical behavior of the set of normality and the Julia set. another reason for studying these semigroups is that certain polynomial semigroups are closely connected to moduli spaces of discrete Mobius groups. The dynamical systems point of view offers a new tool for studying and understanding such moduli spaces. A second line of investigation involves the question of characterizing a convergence group on the circle in terms of fixed points, as well as with the problem of determining under what circumstances a convergence group in the plane is topologically conjugate to a Mobius group. In dimensions two and higher, there are a considerable number of convergence groups, not all of which are conjugate to Mobius groups. It is important that efforts be made to determine the constraints necessary to guarantee that a group have the same topological properties as a Mobius group. Classical function theory has introduced geometric techniques which have proved to have far wider applicability than first imagined. The subject exploits the interplay between geometric interpretation with powerful analytic techniques to yield some of the most complete mathematical results in the field of analysis. In this project, variational techniques and geometric reasoning are expected to lead to results of valuable physical significance. ***