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Institut de Mathématiques de Bourgogne - UMR 5584 CNRS
Université de Bourgogne
9 avenue Alain Savary
21078 DIJON Cedex
Qualitative Theory of Dynamical Systems: Basic ideas about recurrent sets.
The theory of Dynamical Systems is a branch of mathematics that studies the movement. Among the several approaches to the concept of movement, the so called Deterministic Dynamical Systems is the most classical one, whose beginning is usually associated to the foundational works of Newton and Leibnitz during the seventeenth century. In face of the impossibility of solving differential equations, a new approach was developed in the second half of the nineteenth century, mainly due to the works of Poincaré, and which is known as the Qualitative Theory of Dynamical Systems.
The purpose of this talk is to present one of the most basic concepts of this theory, the concept of Recurrence. We will explain some classical examples in dimension one, making special emphasis in the linear expanding maps of the interval, which will serve as toy model for Chaos.
We hope to complement this talk with a second one about Anosov Dynamical Systems (and general notions about Chaos Theory), for which the present talk will serve as previous knowledge.