Stretching and shrinking: 85 years of the Hopf argument for ergodicity

Time:Wednesday, February 7, 2024 9:00–10:30 am ET 22:00-23:30 (Beijing Time)

Venue: Zoom Webinar Registration

Speaker:Prof. Amie Wilkinson University of Chicago


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.


Amie Wilkinson is an American mathematician and Professor of Mathematics at the University of Chicago. Her research topics include smooth dynamical systems, ergodic theory, chaos theory, and semisimple Lie groups. Wilkinson, in collaboration with Christian Bonatti and Sylvain Crovisier, partially resolved the twelfth problem on Stephen Smale's list of mathematical problems for the 21st Century.

Wilkinson was named a fellow of the American Mathematical Society (AMS) in 2014. She was elected to the Academia Europaea in 2019 and the American Academy of Arts and Sciences in 2021. In 2020, she received the Levi L. Conant Prize of the AMS for her overview article on the modern theory of Lyapunov exponents and their applications to diverse areas of dynamical systems and mathematical physics.

DATEFebruary 7, 2024
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