Celebratio Mathematica

I think I first met Bob at a Sum­mer In­sti­tute of the Amer­ic­an Math­em­at­ic­al So­ci­ety held in Berke­ley in the sum­mer of 1968. I was a gradu­ate stu­dent fin­ish­ing my dis­ser­ta­tion and he was a pro­fess­or at North­west­ern Uni­versity (NU) and lec­tured on his work on one-di­men­sion­al at­tract­ors. At the time I had no ink­ling of the in­flu­ence he was to have on my life and work. He would be in­flu­en­tial in my ca­reer, my math­em­at­ic­al tastes, and would in­tro­duce me to my fu­ture wife.

The sum­mer of 1968 was a time when the so-called Smale school of dy­nam­ics was just emer­ging and Bob was one of the lead­ing play­ers — per­haps the most prom­in­ent of those with no dir­ect con­nec­tion to Berke­ley.

In the sub­sequent few years, Bob was al­most single-handedly re­spons­ible for the cre­ation of a strong dy­nam­ics group at North­west­ern, a group which con­tin­ues to this day. He first brought Clark Robin­son and Shel­don Ne­w­house, and a year later I was hired at North­west­ern. The group that vis­ited dur­ing the next few years in­cluded Paul Blan­chard, Bob Devaney, Bruce Kit­chens, Den­nis Pix­ton, S. Shah­shahani, and Lai-Sang Young, as well as sev­er­al oth­er postdocs.

Early in the 1970s Bob and I were both on leave and spent some time at the In­sti­tut des Hautes Études Sci­en­ti­fiques (IHES). Bob was to serve as the ex­tern­al ex­am­iner on the Ph.D. de­fense of An­thony Man­ning, but could not re­turn from a climb­ing trip in time. Des­pite my very ju­ni­or status I was draf­ted as his emer­gency re­place­ment.

Bob and I had con­sid­er­able over­lap in our math­em­at­ic­al in­terests and it was nat­ur­al for us to work to­geth­er on some prob­lems. A cent­ral mo­tif of our joint work was the re­la­tion between dy­nam­ics and to­po­logy: what dy­nam­ic­al prop­er­ties are forced by to­po­lo­gic­al re­stric­tions or con­versely when does dy­nam­ic­al com­plex­ity im­ply the ex­ist­ence of to­po­lo­gic­al com­plex­ity. A good ex­ample of this is our joint pa­per [2] in which we proved that a smooth flow on $S^3$ which has pos­it­ive to­po­lo­gic­al en­tropy must have closed or­bits rep­res­ent­ing in­fin­itely many knot types. An­oth­er joint work of which I am proud was [1] where we con­struc­ted an ex­ample of an Anosov flow which is not trans­it­ive, and in­deed has both a hy­per­bol­ic at­tract­or and a hy­per­bol­ic re­peller. It had been con­jec­tured that all Anosov flows and dif­feo­morph­isms were trans­it­ive. This ex­ample dis­proved the con­jec­ture for flows, but the ques­tion for dif­feo­morph­isms re­mains open.

I learned a lot of math­em­at­ics from Bob. In par­tic­u­lar I learned the im­port­ance of solen­oids (these are in­ad­equately ap­pre­ci­ated math­em­at­ic­al ob­jects which oc­cur nat­ur­ally in dy­nam­ics). I also learned much about the sym­bol­ic dy­nam­ics of sub­shifts of fi­nite type. He in­tro­duced the no­tion of shift equi­val­ence and con­jec­tured that to­po­lo­gic­al con­jugacy for ir­re­du­cible sub­shifts of fi­nite type would be char­ac­ter­ized by shift equi­val­ence of their as­so­ci­ated matrices. Two morph­isms $A$ and $B$ in a cat­egory are said to be shift equi­val­ent provided there ex­ist morph­isms $R$ and $S$ and pos­it­ive in­teger $n$, such that $$A^n=SR,B^n=RS,RA=BR,\text{and}BS=SA.$$ As he men­tions, the con­jec­ture has turned out to be false, but a stronger (though less tract­able) no­tion of strong shift equi­val­ence does char­ac­ter­ize to­po­lo­gic­al con­jugacy of these sys­tems. From con­ver­sa­tions with Bob I gained an ap­pre­ci­ation of the im­port­ance of shift equi­val­ence in many cat­egor­ies, not just pos­it­ive in­teg­ral matrices.

It was a sad day for me when Bob left North­west­ern for Texas. His most math­em­at­ic­ally pro­duct­ive years were spent at NU and his pres­ence was missed. However, he left the leg­acy of a strong dy­nam­ics group in the math­em­at­ics de­part­ment and of a ma­jor role as one of the founders of the Mid­w­est Dy­nam­ic­al Sys­tems Sem­in­ar. Of course his most last­ing leg­acy will be his many con­tri­bu­tions to math­em­at­ics.

John Franks is the Henry S. Noyes Pro­fess­or Emer­it­us in Math­em­at­ics at North­west­ern Uni­versity.