The investigators will continue their research involving measures of various kinds-Hausdorff, packing, Gibbs state, etc.; associated functions; and associated dimensions. These notions will be studied within the context of various dynamical systems as applied to Julia sets for example or to characterization of Jordan curves as repellers. They will use these techniques with regard to certain geometric objects and constructions, e.g. self-affine sets and self-- similar objects. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics" can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way, ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.
