Discussion

Sheldon and I have been thinking about this discussion for about 45 minutes. It’s not well prepared, but we agreed a moment ago that I would talk a little bit about the work of Rufus Bowen, John Franks, John Guckenheimer, and others on zeta functions, since that was something that Jacob did not have a chance to talk about. It was, of course, one important thing about the 1967 paper which many of us were working hard on in 1966, because it came out in preprint. It was very much an outline. In it, Smale did things that he’s done so many times that have been so helpful to us. He’s laid out ideas with strong statements, conjectures if you wish, which he didn’t feel all that secure about, but he’s brave enough to make the conjectures, so we got to play with them. Two kinds of zeta functions were defined. Some proofs were given, some computations were made. I remember very well correcting a slight error of his in this paper, and pointing it out, and of course when it came out, not only was the error corrected but he used my own notations…. Anyway, he was brave enough to conjecture something that took years to prove by John Guckenheimer—first Mañé—the rationality of the zeta functions.