Abstract:

Anosov diffeomorphisms on closed manifolds (ie, compact without boundary) have been the subject of extensive study over the past half-century.  The analogous objects (diffeomorphisms with a global hyperbolic structure) in a non-compact setting have hardly been studied at all.  In 1970, Warren White constructed a Riemannian metric on the plane for which the translation (x,y)->(x, y+1) is globally hyperbolic, thus showing that the notion of a hyperbolic structure is very dependent on the uniform structure of the metric being used.  Pedro Mendes showed that certain properties of Anosov diffeomorphisms (specifically structural stability) carry over to the case of the plane, and conjectured that every Anosov diffeomorphism of the plane is equivalent ( a strengthening of topological conjugacy) to either Warren White's example or to the standard linear hyperbolic diffeomorphism.  With Jorge Groisman, we have obtained a counterexample to this conjecture by constructing a complete metric on certain subsets of the plane for which the linear hyperbolic map is hyperbolic.  I plan to present some of these results, especially our example, and to speculate on possible "corrections" to Mendes' conjecture.