ABSTRACT Lakic This project addresses problems concerning metric and analytic properties of infinite dimensional Teichmuller spaces. Properties of some recently introduced new Teichmuller spaces will be explored. These new spaces appear to have interesting applications to the study of complex dynamics. Three closely related areas will be considered in this project. The first concerns the classical Teichmuller theory, and it is based on classifying the automorphisms of infinite dimensional Teichmuller spaces and on studying uniquely extremal Beltrami coefficients. The second area focuses on two recently introduced Teichmuller spaces, the Teichmuller space with asymptotic conformal equivalence and the Teichmuller space of a closed set in the plane. Finally, some applications will be considered. A special emphasis will be placed on studying the invariant line fields on a Julia set, and on exploring the smoothness properties of the mapping by heights for quadratic differentials in the disk. Recently, scientists have been able to apply several new analytic and geometric methods to a number of important nonlinear problems in dynamics, ranging from physics and chemistry to ecology and economics. In particular, the methods of Teichmuller theory have contributed to the depth and scopes of research in complex dynamics and renormalization. A special emphasis will be placed on studying the Julia set, that appears in the iteration theory of holomorphic functions that gave birth to the Mandelbrot set and to fractals.
