CRM COLLOQUIUM 2026
Susanna Terraccini
Università di Torino
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ABSTRACT
On the Birkhoff conjecture for Kepler billiards
We investigate the integrability of Kepler billiards-mechanical billiard systems in which a particle moves under the influence of a Keplerian potential and reflects elastically at the boundary of a strictly convex planar domain. Our main result establishes that, except possibly for one location of the gravitational center, analytic integrability at high energies occurs only when the domain is an ellipse and the center is placed at one of its foci. This provides a partial affirmative answer to a Keplerian analogue of the classical Birkhoff-Poritsky Conjecture.
Our approach is based on the construction of symbolic dynamics arising from chaotic subsystems that emerge in the high-energy regime. Depending on the geometric configuration of the boundary and the location of the attraction center, we construct three types of symbolic dynamics by shadowing chains of punctured Birkhoff-type trajectories. These constructions yield subsystems conjugated to Bernoulli shifts, implying positive topological entropy and precluding analytic integrability.
We further analyze the notion of focal points of the second kind, showing that in real-analytic, non-elliptic domains there can be at most one such point, while ellipses are the only domains admitting two-coinciding with their classical foci. Finally, we demonstrate the existence of an infinite-dimensional family of non-elliptic domains possessing a focal point of the second kind, and conclude with numerical simulations illustrating chaotic behavior in such cases.
Based on a joint work with Stefano Baranzini, Vivina L. Barutello and Irene De Blasi
SPEAKER
Susanna Terracini
Università di Torino
Terracini was born in South London. She earned a laurea in 1986 in mathematics at the University of Turin, supervised by Fulvia Skof. She completed her Ph.D. at the International School for Advanced Studies in 1990. Her dissertation, Periodic Solutions to Singular Newtonian Systems, was supervised by Ivar Ekeland and Sergio Solimini. She was a researcher at Paris Dauphine University from 1988 to 1989, and became a faculty member at the Polytechnic University of Milan in 1990. In 2001 she became a full professor at the University of Milano-Bicocca, and in 2012 she returned to Turin as a professor.
One of Terracini’s papers on the n-body problem was selected for a featured review in Mathematical Reviews. She was the winner of the 2002 Vinti Prize, a prize of the Italian Mathematical Union for young researchers in mathematical analysis. In 2007 she won the Bruno Finzi Prize of the Istituto Lombardo Accademia di Scienze e Lettere. In 2020 she was awarded the Schauder Medal from the Juliusz P. Schauder Center for Nonlinear Studies at the Nicolaus Copernicus University in Toruń, Poland.
SCHEDULE
| Time | Tuesday 14 July 2026 |
|---|---|
| CRM Colloquium | |
| 16:00→ 16:10 | Welcoming |
| 16:10→ 17:10 |
On the Birkhoff conjecture for Kepler billiards
Susanna Terracini
Università di Torino (and member of the SAB)
|
| Inauguration of the CRM Directors Gallery | |
| 17:15 | Gallery of Directors Opening |
| 17:30 | A toast of cava |
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For inquiries about this event, please contact the Head of Scientific Events, Ms Núria Hernández, at nhernandez@crm.cat
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