Renormalization and rigidity of circle maps with a singularity and the spectrum of the Schroedinger operators over them

Sasa Kocic

I will discuss the renormalization and rigidity of circle maps a singularity and some recent results concerning the spectrum of the Schroedinger operators over them. In particular, we prove that, for any C^{2+ε}-smooth circle map with a break (or any C^3-smooth critical circle map) T, with an irrational rotation number, and an invariant measure μ, for μ a.e. x, and for every Holder-continuous potential V, in a region of the Lyapunov exponent, the spectrum of the Shroedinger operator H(T,V,x) is purely singular continuous. I will also state a sharp transition result for maps with breaks and explain the role that the geometry of dynamical partitions plays in these problems.

http://scgp.stonybrook.edu/video_portal/video.php?id=4702

Simons- Program: Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory