Dinâmica simbólica da ferradura de Smale

Abstract

The Smale horseshoe is a map built by taking a square that is stretched in a rectangle on and then folded into a horseshoe shape. This horseshoe is overlapped the original square and the process is repeated indefinitely, providing a geometric shape capable of showing the presence of unpredictability in discrete dynamics. This work has as general objective to understand the dynamics associated with horseshoe and its relation with the symbolic dynamics. At each iteration of the horseshoe map, several points are removed from the square. These points have a simpler dynamic. The points whose orbits remain in the square form a Cantor like set and present a chaotic behavior. The dynamics of the family of quadratic functions Fµ = µx(1 − x), with µ > 2 + √ 5, is very similar to the map horseshoe and is also developed in this work. Its analysis is done through a topological conjugation with the shift map in the metric space of the infinite sequences of terms 0 and 1.