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Dynamical properties of some classes of interval maps

Abstract

Poincare's work was the forerunner of modern theory of dynamical systems. Since its introduction, this theory has grown and matured, becoming an important and much studied area of mathematics. The main objective of this project is to deepen the knowledge in the following areas of dynamical systems:1 - Renormalization Theory and Cherry flows.2 -Rigidity for one-dimensional dynamical systems. In [16] we study some dissipative Lorenz maps of the interval, which are maps of the interval having a discontinuity point and positive derivative (and uniformly) smaller than one at every point of your domain. Interested in the dynamics of these maps, we study periodic orbits, renormalizations and the invariant minimal singular set when there is no periodic orbit. In a specific set of these maps we prove the existence of a lamination corresponding to the innitely renormalizable maps, as well as the regularity of the leaves of that lamination, in the analytical case. We also have been able to study the regularity of the conjugations and the holonomy maps of the lamination when the maps have no criticality. In this context we aim to: obtain better properties in the as, for instance, to study the generalized Hausdor dimension of the minimal singular set; to extend these results for the case where maps have Lorenz singularity; to study gap maps presenting a plateau in one of its branches; to study gap maps with more than one gap and obtain results similar to [16] for these gap maps with more than one gap; to study the dynamics of gap maps in the non uniformly contracting case. (AU)

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