Zhiqiang Li
(Peking University, China)
Misha Lyubich
(Stony Brook University, USA)
Michael
Yampolsky (University of Toronto, Canada)
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 of partially hyperbolic symplectomorphisms.
On the center-bunched locus, for , previous work of Dolgopyat–Wilkinson
and Burns–Wilkinson gives a -dense set of -stably ergodic maps. Beyond the center-bunched
locus, no open-dense result is known. Earlier, Avila, Bochi,
and I proved that ergodicity is -generic in . I will describe recent work in progress with Avila and Crovisier aiming for a -open (for r sufficiently large) and -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.
Zoom Meeting ID: 827 8743 3327
Passcode: 692502
Speaker 1
Raphaël Krikorian (Institut Polytechnique De Paris, France):
Title: Exotic rotation domains and Herman rings for
quadratic Hénon maps
Abstract: Quadratic Hénon
maps are polynomial automorphism of $\mathbb{C}^2$ of
the form $h:(x,y)\mapsto (\lambda^{1/2}(x^2+c)-\lambda y,x)$.
They have constant Jacobian equal to $\lambda$ and they admit two fixed points.
If $\lambda$ is on the unit circle (one says the map $h$ is conservative) these
fixed points can be elliptic or hyperbolic. In the elliptic case, a simple
application of Siegel Theorem shows (under a Diophantine assumption) that $h$
admits many quasi-periodic orbits with two frequencies in the neighborhood of
its fixed points. Surprisingly, in some hyperbolic cases, S. Ushiki observed some years ago what seems to be
quasi-periodic orbits though no Siegel disks exist. I will explain why this is
the case. This theoretical framework also predicts and mathematically proves,
in the dissipative case ($\lambda$ of module less than 1), the existence of (attractive)
Herman rings. These Herman rings, which were not observed before, can be
produced in numerical experiments.
Speaker 2
Bertrand Deroin (CNRS -
CYU, France):
Title: Towards a Structural Stability Theory for
Holomorphic Foliations on Algebraic Complex Surfaces
Abstract: I'll review work done in collaboration
with Aurélien Alvarez, aiming at developing a theory
of structural stability for holomorphic foliations on compact complex surfaces.
Important new examples are the Jouanolou foliations
of the complex projective plane, and the fundamental properties they satisfy
allow us to define a more general family of foliations on arbitrary algebraic
surfaces, which we call Jouanolou-type foliations. I
will present these conditions, as well as some of their properties, and, time
permitting, I will state a number of conjectures.
Speaker 3 Giovanni
Forni (CYU, France - University of Maryland, USA)
Title: On the dynamics of billiards in polygons
Abstract: In this talk we will survey several
results on the dynamics of billiards in polygons. These include
results on their ergodic theory (weak mixing), KAM-type results on stability of
invariant surfaces of rational billiards under perturbations and existence of
periodic orbits. Part of the work is in collaboration with F. Arana Herrera and
J. Chaika.
Last Updated: 5/27/2026