Author: Daniel Tsodikovich

Beatrice Langella (Università di Pisa)

Abstract: Water waves equations describe the time evolution of an incompressible fluid evolving according to Euler equation, in presence of free boundary. We consider water waves equations in the two-dimensional periodic setting, in presence of gravity, constant nonzero vorticity and infinite depth.
Under suitable resonance assumptions on the value of the vorticity, we show that this system admits smooth small-amplitude solutions whose high Sobolev norms undergo arbitrary large growth, whereas their lower Sobolev norms remain small, thereby exhibiting a genuine transfer of energy toward high frequencies.
This yields the first rigorous construction of weakly turbulent solutions for a quasilinear hydrodynamic wave system, in a regime where the flow remains smooth. Moreover, the growth occurs simultaneously in the free surface and in the vertical component of the velocity at the interface, showing that the instability involves the full hydrodynamic evolution.The proof relies on a new mechanism for generating energy cascades in quasilinear dispersive PDEs with sublinear dispersion and a nonlinear transport structure.
This talk is based on a joint work with A. Maspero, F. Murgante and S. Terracina.

Lucas Dahinden (Utrecht University)

Abstract: Topological entropy is a numerical invariant of dynamical systems that measures chaotic behavior. Entropy is unstable: There are entropic dynamical systems that can be smoothly perturbed to zero entropy systems. In this talk, we show that in the case of geodesic flows, sometimes a geometric feature prevents this loss. An example of such a feature is a closed contractible geodesic on a two-dimensional torus. We will also investigate general manifolds with a ‘long retractable neck’.In these situations, entropy persists even under C^0 perturbation of the metric. This is (to me) surprising, as not even the geodesic flow is continuous under such perturbations.This is joint work with Marcelo Alves, Matthias Meiwes and Louis Merlin.

Alfonso Sorrentino (University of Rome Tor Vergata)

Abstract: In this talk, I will present a stochastic homogenization result for first-order Hamilton-Jacobi equations on Riemannian manifolds within a stationary ergodic random environment. The setting involves a finitely generated abelian group of isometries acting freely and cocompactly on the manifold, together with a family of Hamiltonians parametrized by an ergodic probability space. Under standard assumptions – including strict convexity and superlinear coercivity in the momentum variable -I will show that as the scaling parameter tends to zero, the viscosity solutions to the rescaled Hamilton-Jacobi equation converge almost surely and locally uniformly to the solution of a deterministic homogenized equation posed on a Euclidean space of dimension d, where d corresponds to the rank of the acting group. This is based on joint work with Marco Pozza.

Stefano Baranzini (Università San Raffaele Roma)

In this talk I will discuss recent pointwise spectral rigidity results for several billiard systems, including Birkhoff billiards and symplectic billiards. The main theme is that, in many cases, a single value of Mather’s beta-function is enough to determine whether the domain is a disk (or an ellipse in the affine-invariant setting of symplectic billiards). This provides a “one-frequency” analogue of the classical question “Can one hear the shape of a billiard?”

The results are based on isoperimetric-type inequalities comparing the beta-function of a convex domain with that of a disk having the same perimeter or area, depending on the model. I will explain how equality in these inequalities leads to rigidity, and how this viewpoint gives a dynamical reinterpretation of classical geometric inequalities for extremal polygons. Time permitting, I will also discuss a contrasting phenomenon for outer billiards, where such rigidity fails.
This is a joint work with M. Bialy and A. Sorrentino.

Mar Giralt (Observatoire de Paris)

Abstract: In the Restricted Planar Circular 3-Body Problem, which models the dynamics of a massless body influenced by two massive bodies in circular orbits, the Lagrange point L3 is a saddle-center critical point. We explore the family of periodic orbits (close enough) to L3 and prove that these orbits intersect transversally, leading to chaotic dynamics. Furthermore, we identify a generic unfolding of a quadratic homoclinic tangency, which gives rise to Newhouse domains. The talk is based on a joint work with Inma Baldomá and Marcel Guardia.

Tobias Jäger (University of Jena)

We give an overview of some general concepts and methods for the 
construction of irregular model sets, obtained via Meyer’s cut and 
project schemes (CPS), with prescribed additional dynamical 
properties. This allows to  roduce specific examples at the very 
opposite ends of dynamical complexity: On the one hand, one may obtain 
irregular model sets with positive entropy [1]. On the other hand, it 
is possible to construct model sets such that almost all fibres over 
the torus parametrisation have a given finite cardinality k ∈ N [2,3]. 
Some intermeditate cases can be realised as well, for instance model 
sets whose fibres are almost surely infinite, but which only allow a 
finite number of ergodic invariant measures.

On the technical level, a special role is played by certain windows 
with a very particular symmetry property, which we refer to as perfect 
self-similarity (with respect to the window of the CPS) The 
above-mentioned examples can all be realised in the Euklidean setting. 
However, the methods can also be applied to the construction of 
Toeplitz sequences or Toeplitz arrays over countable and residually 
finite groups.

The presented results are contained in a series of works, including 
joint work with Jamal Drewlo, Daniel Lenz, Gabriel Fuhrmann, Christian 
Oertel, Maria Isabel Cortez and Jaime Gomez.

References:

[1] Model sets with positive entropy in Euclidean cut and project 
schemes. T. Jäger, D. Lenz and C. Oertel. Ann. Sci. École Norm. Sup. 
52:1073–1106, 2019.
[2] Irregular model sets and tame dynamics. G. Fuhrmann, E. Glasner, 
T. Jäger and C. Oertel. Trans. Amer. Math. Soc. 374(5):3703-3734,2021.
[3] Toeplitz arrays obtained via cut and project schemes. M.I. Cortez, 
J. Drewlo, J. Gomez and T. Jäger. In preparation.
[4] A survey on model sets. J. Drewlo, T. Jäger and D. Lenz. In preparation.

Kosma Kasprzak (Jagiellonian University)

Abstract: One of the main trends in sparse ergodic theory is examining ergodic averages on a dynamical system $(X, T)$ along sparse orbits of the form $(T^{P(n)}(x))$, for polynomials $P$. We will focus on the cases where the sequence of such averages converges for every starting point, and in particular when each polynomial orbit is equidistributed in $X$. These are very delicate properties, and not many such systems are known — we lack examples with mixing properties or ones exhibiting different behavior for different choices of $P$.
I will introduce the known approaches and discuss the above limitations, before focusing on an original result stating that the desired equidistribution holds under a rigidity assumption. Here by rigidity we mean the existence of a sequence $(T^{q_n})$ of iterates of $T$ converging to the identity map on $X$ — in our case we need strong quantitative bounds on the rate of convergence, which however still allow for weakly mixing examples.

Marco Lenci (University of Bologna)

Abstract: We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the pretense that the boundary at
any collision point is either horizontal or vertical (relative to a predetermined direction representing gravity). These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. The phenomenon of internal waves in a fluid occurs in many situations and has been intensively studied by physicists. One of the first experiments, which became paradigmatic, was done in a container shaped like a rectangular trapezoid (with some thickness).
For a class of tables including rectangular trapezoids, we prove that the dynamics has only three asymptotic regimes: (1) there exist a global attractor and a global repellor, which are periodic and might coincide; (2) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories; (3) all trajectories are dense (that is, the system is minimal). If time permits, we will also discuss the prominent case where the table is an actual trapezoid, studying the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) has positive measure (giving a rigorous proof of the existence of Arnold tongues for internal-wave billiards), whereas the sets for (2) and (3) are non-empty but have measure zero.
Joint work with C. Bonanno and G. Cristadoro.

Daniel Tsodikovich (ISTA)

Abstract: The Toda lattice is a completely integrable mechanical Hamiltonian system, in which particles interact according to nearest neighbor interaction law. We study deformations of the Toda Hamiltonian, and show that within certain classes of deformations, the integrability of the Toda lattice is rigid. This is a joint work with Vadim Kalsohin and Illya Koval.

Jaelin Kim (Renyi Institute)

Abstract:  In the finite graph, the maximal entropy random walk is well understood. In the ergodic-theoretic view point, it is the invariant measure of maximal entropy of the subshift of finite type over vertex set, induced by adjacency matrix. In this talk, we will introduce a generalization of MERW on infinite graphs, which we call a uniform random walk. By observing its behavior on graphs with weighted loops, we will see it gives meaningful notion even in the transient case.