by John Franks
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 \[ A^n = SR,\quad B^n = RS, \quad RA = BR, \quad \text{and} \quad BS = SA. \] 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.
