Properties of invariant sets in certain two-dimensional dynamical systems : Renormalization and beyond

Sammanfattning: This thesis consists of an introduction and four research papers concerning dynamical systems, focusing on renormalization in two dimensions.Paper I (joint with Jordi-Lluís Figueras) studies a generalization of the anti-integrable limit of skew-product systems based on a Frenkel-Kontorova model. It is shown that under certain regularity conditions the orbits of such a system has a Cantor set structure and the existence of a non-smooth folding bifurcation is deduced.Paper II studies the invariant Cantor sets of period doubling type of infinitely renormalizable area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point. It is shown that for such maps the invariant Cantor set is always contained in a Lipschitz curve but never in a smooth (meaning C1) curve.Paper III (joint with Denis Gaidashev) continues the study of the invariant Cantor sets of infinitely renormalizable area-preserving maps, focusing on their geometry. It is shown that there is always a positive measure of unbounded geometry. Additionally, some results giving control over the average expansion of vectors by the renormalization microscope are also providedFinally paper IV (joint with Denis Gaidashev) considers renormalization of two-dimensional perturbations of almost commuting pairs of holomorphic maps. A renormalization operator is developed and statements about the hyperbolicity, universality and non-rigidity of this operator are proven.