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4:00 pm Wednesday, January 18, 2012 Geometry-Analysis Seminar: Dynamics of the Fibonacci trace map, geometry of dynamical invariants and applications to spectral theory of 1D quasicrystalsby
William Yessen (UC Irvine) in HB 227- Since the discovery of quasicrystals in the early 80's by Schechtman et. al., quasiperiodic models in mathematical physics have gained the status of an independent research field. In addition to spectral theory, a few (seemingly unrelated at first glance) tools from other areas have been successfully applied. In particular, a fascinating connection between smooth dynamical systems and spectral theory of 1D quasicrystals was established during the early years of development. After motivating the problem in terms of spectral theory of certain so-called quasiperiodic Hamiltonians, we shall discuss briefly a relation with the so-called Fibonacci trace map---a three-dimensional polynomial map on the Euclidean three-space---and then in some detail dynamics of the Fibonacci trace map and geometry of dynamical invariants (invariant sets, stable and unstable manifolds, etc.). We'll close the loop by applying this knowledge to the spectral problem, consequently establishing the following: 1) topology of the spectrum, 2) measure-theoretic properties, 3) fractal-dimensional properties. We'll conclude with some open problems and conjectures.
Submitted by damanik@rice.edu |