On Topology: On Smale's Horseshoe

"Instead of a state of nature evolving according to a mathematical fomula, the evolution is given geometrically. The full advantage of the geometrical point of view is beginning to appear. The more traditional way of dealing with dynamics was with the use of algebraic expressions. But a description given by formulae would be cumbersome. That form of description wouldn't have led me to insights or to perceptive analysis. My background as a topologist, trained to bend objects like squares, helped to make it possible to see the horseshoe." (Steven Smale, "What is chaos?", 1990)

"The horseshoe is a natural consequence of a geometrical way of looking at the equations of Cartwright-Littlewood and Levinson. It helps understand the mechanism of chaos, and explain the widespread unpredictability in dynamics." (Steven Smale, "What is chaos?", 1990)

"This construction, the horseshoe, has some consequences. First, it yields the fact that homoclinic points do exist and gives a direct construction of them. Second, one obtains such a useful analysis of a general transversal homoclinic point that many properties follow, including sensitive dependence on initial conditions - 'a large number', anyway. Third, one can prove robustness of the horseshoe in a strong global sense structural stability)." (Steven Smale, "What is chaos?", 1990)

"Here is the result of the horseshoe analysis that I found on that Copacabana beach. Consider all the points which, under the horseshoe mapping, stay in the square, i.e., don't drift out of our field of vision. These 'visual motions' correspond precisely to the set of all coin-flipping experiments! This discovery demonstrates the occurrence of unpredictability in fully non-linear motion and gives a mechanism of how determinism produces uncertainty."(Steven Smale, "What is chaos?", 1990)

"At least in my own case, understanding mathematics doesn't come from reading or even listening. It comes from rethinking what I see or hear. I must redo the mathematics in the context of my particular background. And that background consists of many threads, some strong, some weak, some algebraic, some visual. My background is stronger in geometric analysis, but following a sequence of formulae gives me trouble. I tend to be slower than most mathematicians to understand an argument. The mathematical literature is useful in that it provides clues, and one can often use these clues to put together a cogent picture. When I have reorganized the mathematics in my own terms, then I feel an understanding, not before." (Stephen Smale, "Finding a Horseshoe on the Beaches of Rio", 2000)

"The Smale's horseshoe is the classical example of a structurally stable chaotic system: Its dynamical properties do not change under small perturbations, such as changes in control parameters. This is due to the horseshoe map being hyperbolic (i.e., the stable and unstable manifolds are transverse at each point of the invariant set)." (Robert Gilmore & Marc Lefranc, "TheTopologyof Chaos: Alice in Stretch and Squeezeland", 2002)