Teichmuller spaces of closed sets have many applications to dynamics and hyperbolic geometry. The recently developed asymptotic Teichmuller spaces, for instance, serve as parameter spaces in the study of dynamical systems. Modern dynamical system theory has had a number of important applications, ranging from the life sciences and physics to economics. Several complicated biological phenomena (the rate of change in the population of a single species, to cite a concrete example) are modelled by nonlinear dynamical systems, making it imperative to understand the long-term behavior of these mathematical systems if knowledge of the corresponding bio-system is to be advanced.