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4:00 pm Wednesday, September 4, 2013 Geometry-Analysis Seminar: Polynomial Trace Maps and some of their Applicationsby
William Yessen (Rice University) in HB 227- In this largely expository talk, we shall discuss a fascinating family of polynomial maps, acting on the three-dimensional Euclidean space (complex or real). These maps have been studied mainly in two contexts: (1) spectral theory of a family of 1-dimensional discrete quasi-periodic Schroedinger operators, and 2) as a source of rich dynamical behavior. In the former case, to a given quasi-periodic potential there is a way to associate a trace map in such a way, that the spectrum (as a set) is related to a certain dynamically invariant set of the trace map. Some other quantities, such as the density of states measure, can also be related to some dynamical invariants of the trace map. Regarding (2): trace maps have been shown to exhibit quite rich dynamical behavior, such as Axiom A, partial hyperbolicity and, quite recently, mixed behavior with a large (in the sense of the Hausdorff dimension) chaotic see. The last point is an important one, as it demonstrates that trace maps present perhaps an even simpler example of an analytic symplectic map with such properties than the famous "Taylor-Chirikov standard map" (which so far has been the simplest known example). On the other hand, there are a few major unresolved conjectures about the Taylor-Chirikov map which can now be reformulated in terms of trace maps.
Submitted by damanik@rice.edu |