Executive Summary : | The one-dimensional theory focuses on the geometric rigidity of attractors, with renormalization being the main technique used to study the microscopic geometric properties of these attractors. Introduced by P. Coullet, C.P. Tresser, and M.J. Feigenbaum, the method was initially used to study dynamics at the accumulation of period doubling and the boundary of chaos in system space. The attractors of transitioning maps have a unique property, being Cantor sets. On arbitrarily small scale, the attractor can be identified with a rescaled version of another one-dimensional map. This allows for the introduction of an operator on the set of one-dimensional maps at transition, acting as a microscope, allowing infinitely many applications to study dynamics. The project aims to construct the renormalization operator to study the geometric properties of multimodal maps with low smoothness and describe the geometrical and topological behavior of infinitely renormalizable multimodal maps. |