Research Interests

My mathematical work began with the study of hyperbolic dynamical systems (i.e., systems whose long-range behavior exhibits exponential growth and decay of distances along trajectories in complementary directions). They play a central role in dynamics, and once it was thought that every system could be approximated by a hyperbolic one, an idea that soon collapsed except for special sets of systems like the gradient ones. Still, I have proved with Smale that many such systems are dynamically stable when slightly perturbed. Moreover, we conjectured that hyperbolicity is essentially the same as being dynamically stable, which in subsequent decades was proved to be true. I have also conjectured that near every system there is one for which most events will have only finitely many choices where to evolve in the future attractors; the attractors are now required to be only stochastically stable. Several mathematicians proved that the conjecture is true for unimodal maps of the interval. There are also important results on homoclinic (same past and future) bifurcations.

Membership Type

International Member

Election Year


Primary Section

Section 11: Mathematics

Secondary Section

Section 32: Applied Mathematical Sciences