Celebratio Mathematica

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.