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.