Abstract:  The early 20th century witnessed an explosion of activity, much of it centered at Harvard, on rigorizing the property of ergodicity first proposed by Boltzmann in his 1898  Ergodic Hypothesis for ideal gases. Earlier, in the 1880’s, Henri Poincaré and Felix Klein had also initiated a study of discrete groups of hyperbolic isometries. The geodesics in hyperbolic manifolds were discovered to carry a rich structure, first investigated from a topological perspective by Emil Artin and Marston Morse.  The time was ripe to investigate geodesics in hyperbolic manifolds from an ergodic theoretic (i.e., statistical) perspective, and indeed Gustav Hedlund proved in 1934 that the geodesic flow for closed hyperbolic surfaces is ergodic.

In 1939, Eberhard Hopf published a proof of the ergodicity of geodesic flows for negatively curved surfaces containing a novel method, now known as the Hopf argument.  The Hopf argument, a “soft” argument for ergodicity of systems with some hyperbolicity (the “stretching and shrinking” in the title) has since seen wide application in geometry, representation theory and dynamics.  I will discuss three results relying on the Hopf argument:

Theorem (E. Hopf, 1939, D. Anosov, 1967): In a closed manifold of negative sectional curvatures, almost every geodesic is directionally equidistributed.

Theorem (G. Mostow, 1968) Let M and N be closed hyperbolic manifolds of dimension at least 3, and let f:M->N be a homotopy equivalence.  Then f is homotopic to a unique isometry.

Theorem (R. Mañé, 1983, A. Avila- S. Crovisier- A.W., 2022) The C^1 generic symplectomorphism of a closed symplectic manifold with positive entropy is ergodic.