by Michael Shub
Bob Williams and I collaborated on two papers: “Future stability is not generic” [1] and “Entropy and stability” [2].
The first of these was written during the Global Analysis Symposium in Berkeley in the summer of 1968. The early conjectures relating stability and genericity in dynamical systems, while formative for the subject, were overly optimistic and were falling rapidly. Examples by Ralph Abraham and Steve Smale. showed that neither structurally stable nor omega stable systems were generic. Future stability was another attempt. Bob and I quickly showed that future stable systems are not generic.
The second paper was rather more important. In 1972, I had made the entropy conjecture that the topological entropy of a \( C^1 \) diffeomorphism of a closed manifold should be greater than or equal to the log of the spectral radius of the induced map on homology. In particular, I conjectured that the special case of axiom A no cycle diffeomorphisms should be true. This is what Bob and I proved. Ruelle and Sullivan were proving the same at roughly the same time. Yomdin proved the general conjecture for \( C^\text{infinity} \) diffeomorphisms about a decade later. The general \( C^1 \) (or \( C^r \) for any finite \( r \) bigger or equal to 1) remains open.
The best thing of all is the friendship that developed between us over the decades. I have stayed with him on several occasions when passing through the Midwest and Texas and he with me when passing through New York. The second paper we have written was written on one of those occasions. Bob was returning from France and either the plane or the boat landed him in New York.
Bob has great joie de vivre especially evident when he is dancing, either at a party in Berkeley or with a escola de samba in Rio.
