Zeta functions in fractal geometry and analysis

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Fractals are mathematical objects which possess detailed structure at all levels of resolution (the middle-1/3 Cantor set is a classical example). They often arise as invariant sets in a dynamical system, but they also have been the subject of extensive study in their own right. A geometric analysis of fractal sets usually proceeds by examining the asymptotics of some measure of their structure as the size scale shrinks to zero. The various notions of dimension (for metric spaces) are all examples of this process. Zeta functions have been used in the analysis of asymptotic properties ever since Riemann first introduced them into the study of the prime number counting function. Zeta functions are powerful tools in this type of analysis and now they are used in many different areas of mathematics outside of number theory (including the study of fractals).******The goal of this project is to significantly extend the use of zeta functions in the study of the geometry of fractal sets and measures. The existing geometric fractal zeta functions encode the Minkowski (box-counting) dimension and Minkowski content of the set via the poles of a suitable meromorphic extension. In some cases, explicit formulae are available which give precise quantitative information about oscillations of the geometry.******The proposed project will extend the geometric reach of fractal zeta functions by defining new zeta functions which are associated with the Hausdorff, Packing and Assouad dimensions and also by defining zeta functions which give "local" information. In addition, the project will explore how these zeta functions are transformed under mappings of the underlying space and define new zeta functions with good mapping properties. The goal here is to create new tools for analyzing projections and slices (intersection with a line) of fractal sets as well as deciding when two fractals are bi-Lipschitz equivalent.******The results of the project will introduce substantial tools for the examination of many geometric measure-theoretic-properties of sets and measures. The fractal zeta functions previously introduced by M. Lapidus and his collaborators have already influenced research in dynamical systems, non-commutative geometry, and theoretical physics (to name just three areas). It seems likely that a broadened class of fractal zeta functions will also find significance in these (and other) areas.