## Morse Inequalities for a Dynamical System in Bulletin of the American Mathematical Society 66 No. 1 pp. 43-49, January 1960 WITH Differentiable Dynamical Systems in Bulletin of the American Mathematical Society 73 No. 6 pp. 747-817, November 1967 WITH Finding a Horseshoe on the Beaches of Rio The Mathematical Intelligencer 20 No. 1 pp. 39-44, Winter 1998 [SMALE’S HORSESHOE & CHAOS THEORY]

1st Edition. THREE FIRST EDITIONS IN ORIGINAL WRAPS DOCUMENTING STEPHEN SMALE’S INVENTION OF THE SMALE HORSESHOE, A MAP THAT BEAUTIFULLY CAPTURES THE MATHEMATICS OF CHAOS THEORY. This important grouping of documents includes Smale’s two seminal papers on his horseshoe as well as a paper by Smale describing how he discovered it. Smale’s 1967 paper alone has been called “a masterpiece of mathematical literature” (Shub, What is a Horseshoe, AMS, 517).

Smale’s Horseshoe is essentially a map that provides a way of looking at chaotic systems that allows for a better understanding of both the mechanism of chaos and its “widespread unpredictability in dynamics” (Scholarpedia). An American mathematician, Smale won the prestigious Field’s Medal in 1966.

The background of Smale’s invention: In 1889, Henri Poincaré worked on a question bothersome to mathematicians. He knew that at least in theory, if he used Newton’s laws of motion and gravitation, he could work out “how a number of objects (say stars, planets and moons) should move due to the gravitational pull they exert on each other. If there are only two bodies involved the answer is straight-forward: they move along curves that are easy to describe (ellipses in the case of a planet orbiting a star). The problem was, what happens if you throw a third object into the mix?” (Freilberger, “Smale’s Chaotic Horseshoe,” 2017).

Here things got tricky. While Poincaré did exceptional work attempting to characterize motions that could result in a system with three bodies, he made a mistake. When he realized his error, he wrote extensively “about what the mistake had revealed to him. What it had revealed was the existence of chaos” (ibid).

Fast forward to 1960 when Smale sat on a beach in Rio thinking over a problem inspired by the mathematics of radio waves. “The complexity of the problem worried him, especially since he had previously conjectured that there was no such thing as chaos” (ibid). A colleague suggested he look at some older papers by the English mathematicians Mary Cartwright and J. L. Littlewood. [Note that we offer some Littlewood papers separately]. Smale thought the papers had curious results that contradicted what he had predicted. He decided to put things into a geometric context so he could understand.

“The result is now know as Smale's horseshoe map. It beautifully encapsulates how chaotic dynamics can arise in mathematics and goes straight to the heart of the problems Poincaré had encountered when thinking about his three-body problem. "Poincaré got a big mess, and [the horseshoe] put order in the mess." (ibid).

Smale's invention, a horseshoe map, functions as follows: A horseshoe map “is any member of a class of chaotic maps of the square into itself…The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe. Most points eventually leave the square under the action of the map. They go to the side caps where they will, under iteration, converge to a fixed point in one of the caps. The points that remain in the square under repeated iteration form a fractal set and are part of the invariant set of the map. The squishing, stretching and folding of the horseshoe map are typical of chaotic systems, but not necessary or even sufficient. In the horseshoe map, the squeezing and stretching are uniform. They compensate each other so that the area of the square does not change. The folding is done neatly, so that the orbits that remain forever in the square can be simply described” (Wikipedia). Item #1261

CONDITION: The backs of the two issues of Bulletin of the American Mathematical Society have been professionally restored. Both issues are in fine and are housed in a custom clamshell case gilt-lettered at the spine and on the front board. The issue of the Intelligencer is in pristine condition but differently sized, is unboxed.

Price: \$700.00