About Limit Sets

Faced with the impossibility of explicitly solving the vast majority of differential equations (such as those governing the movement of planets in space), Henri Poincaré, at the end of the 19th century, proposed to study them from another point of view: that of geometry. From this new approach, a new area of mathematics was created: the area of dynamical systems.

A cornerstone of this theory is that the dynamics are divided into two: the region where the behavior is simple, tame; and the region where the dynamics are complicated, chaotic. The chaotic region is now known as the limit set, and it can have surprising shapes. Although Poincaré's approach was geometric, he himself, in attempting to draw these sets, was faced with such complicated watermarks created by repeating simple rules. It was necessary to develop computers to be able to have very precise representations of these sets; and that took more than a century. Obsessively repeating a simple task, and pushing this repetition to the limit, can bring amazing results. In mathematics, taking a repetition to the limit means performing said task not a thousand, not a hundred thousand times, but repeating it infinitely many times.

Aubin Arroyo

Aubin Arroyo is a researcher at Instituto de Matemáticas, UNAM. He obtained his PhD at Instituto de Matemática Pura e Aplicada (IMPA), Brazil in 2002. Since then he is a researcher at UNAM Institute of Mathematics. His main area of research is Dynamical Systems, but he has also developed work related to Tropical Geometry and Number Theory, among other areas. On the other hand, his work on the visualization of mathematics borders on the fields of art. In particular, his collaboration with the French artist, Jean Michel Othoniel, has been exhibited in various international forums such as the Kirchner Cultural Center (Buenos Aires) and the Museum of Fine Arts of the City of Paris. In Mexico, he has been in charge of the permanent exhibitions at Universum, Museo de las Ciencias, UNAM and at the Morelos State Science Museum; and has created various temporary exhibitions in various national forums.