Renormalization and Quasi-Periodicity

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Renormalization is one of the most powerful tool in asymptotic analysis of dynamical systems. In the proposed research project our aim is to apply the ideology and the tools of renormalization theory to several important problems related to the asymptotic behaviour of quasi-periodic dynamical systems, namely circle diffeomorphisms and circle maps with singularities. We also plan to address problems related to spectral properties of Schrödinger operators with quasi-periodic potentials, and show that two sets of problems are deeply connected. One of the problem in our project is to study renormalization behaviour and rigidity theory for circle maps with multiple break points. Recently we have constructed a renormalization scheme for such maps which has strong symmetry properties. One can say that Rauzy induction, which is usually applied in the case of linear interval exchange transformation, has a very non-trivial counterpart in the setting of Möbius transformations. We intend to show that the map acting on the parameters of the Möbius transformations has strong hyperbolic properties. When hyperbolicity is established one can develop a rich rigidity theory using a well understood tools like distortion estimates etc. Another problem is connected with a parameter dependence for families of circle maps with singularities. It is well known that in the presence of singularities typically the rotation number of a map is rational. We plan to prove that one can define a natural conditional probability distribution on the "irrational" parameter values which exhibits strong universality properties. Namely, the asymptotic properties of such probability distributions depend only on the local structure of singular points, such as the order of the critical points, or the size of a break. We next discuss Schrödinger operators with quasi-periodic potential. It is well-known that in the 1D case there is a transitionfrom absolutely continuous spectrum to the pure point spectrum when the coupling constant in front of the potential is increasing. One can consider a natural family of Schrödinger operators related to the action-minimizing orbits for 2D Standard-type maps. Such orbits are a subject of the Aubry-Mather theory. When action-minimizing orbits belong to KAM invariant curves we expect that the corresponding Schrödinger operator will have a positive measure component of the absolutely-continuous spectrum. We conjecture that after the destruction of invariant curves the spectrum will be pure point. A very difficult but interesting questions arise is the case of critical invariant curves. It seems natural to expect a singular continuous spectrum there. Notice that the renormalization approach can be applied to the study of invariant curves, critical invariant curves, and Cantor-type invariant sets in Aubry-Mather theory, as well as in the analysis of the corresponding Schrodinger operators.