Advances in Dynamics Seminar

The Advances in Dynamics Seminar is an online forum held once per term, dedicated to exploring the latest breakthroughs in the field of dynamical systems.

Organizing Committee

Zhiqiang Li (Peking University, China)

Misha Lyubich (Stony Brook University, USA)
Michael Yampolsky (University of Toronto, Canada)

Date and Time: 28 May 2026, 8am-12pm US Eastern time / 8pm-12am China time

Zoom Meeting ID: 850 3302 8146
Passcode: 331058

Talks:

Speaker 1 Sylvain Crovisier (Université Paris-Saclay, France)
Title: A Closing Lemma for Hénon maps of R2.
Abstract: I will discuss some dynamical properties of mildly dissipative diffeomorphisms of the plane, which includes Hénon maps with jacobian smaller than 1/4. Some years ago we proved with Enrique Pujals a closing lemma: the support of any invariant measure is contained in the closure of the set of periodic points. I will present a stronger version of this closing lemma and state some consequences. In particular any ergodic measure is supported on a periodic orbit, on an homoclinic class or on an odometer. (Collaboration with E. Pujals)

Speaker 2 Andrey Mironov (Novosibisrk State University, Russia)

Title: Integrable billiards inside cones.
Abstract: We study Birkhoff billiards inside cones in $\mathbb{R}^n$. We show that every trajectory inside a cone over a $C^3$ strictly convex closed hypersurface embedded in $\mathbb{R}^{n-1}$ with non-degenerate second fundamental form undergoes only finitely many reflections. Using this result, we prove that the system is both superintegrable and completely integrable. To our knowledge, this provides the first example of an integrable billiard in which the billiard table is neither a quadric nor composed of pieces of quadrics. The results are obtained with Siyao Yin.

Speaker 3 Selim Ghazouani (University College London, UK)
Title: A conjecture about parabolic dynamical systems
Abstract: A parabolic dynamical system is, loosely, a smooth diffeomorphism whose orbits exhibit slow (polynomial) sensitivity to initial conditions. In this talk, I will make the case for the following conjecture: unlike the handful of examples that are well-understood, the generic parabolic dynamical system satisfies the central limit theorem. This case will be based upon numerical experiments and heuristic considerations.

Speaker 4 Amie Wilkinson (University of Chicago, US)

Title: Stable ergodicity of partially hyperbolic symplectomorphisms
Abstract: Symplectomorphisms preserve volume, but their ergodic behavior can be governed by very different mechanisms: Anosov systems are stably ergodic by the Hopf argument, while KAM tori give stable obstructions to ergodicity in high regularity. Partially hyperbolic symplectomorphisms lie between these two regimes. I will discuss the problem of whether stable ergodicity is dense in the space PHS^r of C^r partially hyperbolic symplectomorphisms. On the center-bunched locus, for r>1, previous work of Dolgopyat–Wilkinson and Burns–Wilkinson gives a C^1-dense set of C^1-stably ergodic maps. Beyond the center-bunched locus, no open-dense result is known. Earlier, Avila, Bochi, and I proved that ergodicity is C^1-generic in PHS1. I will describe recent work in progress with Avila and Crovisier aiming for a C^r-open (for r sufficiently large) and C^1-dense version of this generic ergodicity mechanism, using a KAM disk together with a symplectic blender-type construction. A guiding example is an Anosov map coupled to a KAM-type standard map, with the center dynamics not necessarily close to the identity.