Author: Daniel Tsodikovich

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. 

Mathieu Helfter (ISTA)

Abstract: The β-function describes the minimal average action associated with invariant measures of prescribed rotation number. We will discuss rigidity and possible flexibility properties of generalized standard maps defined by analytic potentials exhibiting KAM phenomena on a fixed set of Diophantine rotation numbers.  In particular, we show that, generically, two such potentials have distinct infinite jets of their β-function at a given Diophantine rotation number or that non-degenerate deformations do not preserve the β-function.

Olena Karpel (AGH University of Krakow)

Bratteli diagrams and the dynamical systems arising on their associated path spaces provide a flexible and powerful framework for modeling a broad class of equivalence relations and dynamical phenomena. Originally introduced in the study of AF C∗-algebras, Bratteli diagrams have since become an important tool in the construction of models in measurable, Cantor, and Borel dynamics, where they encode orbit structures, invariant measures, and asymptotic properties of dynamical systems in a combinatorial manner. The talk contributes to the study of invariant measures of Borel dynamical systems that can be modeled using generalized Bratteli diagrams. In this context, we study tail invariant measures on the path spaces of generalized Bratteli diagrams, allowing countably infinite vertex sets at each level. Our main focus is on subdiagrams of generalized Bratteli diagrams
and the problem of extending tail invariant probability measures from vertex and edge subdiagrams to the ambient diagram. We establish necessary and sufficient conditions for the finiteness of such extensions, formulated in terms of incidence matrices and associated stochastic matrices. Several classes of generalized Bratteli diagrams and their subdiagrams are analyzed in detail, including simple, stationary, and bounded size diagrams. We develop constructive, step-by-step procedures for measure extension and for approximating invariant measures by measures supported on suitable subdiagrams. In addition, we provide explicit examples of generalized Bratteli diagrams that admit no probability tail invariant measures, a phenomenon absent for standard Bratteli diagrams with finite vertex sets. Finally, we address convergence questions for sequences of invariant measures arising from approximations by subdiagrams, clarifying the relationship between combinatorial structure and measure-theoretic behavior. The talk is based on results obtained together with S. Bezuglyi, P. Jorgensen, J. Kwiatkowski, T. Raszeja, S. Sanadhya, and M. Wata

Yaniv Ganor (Holon Institute of Technology)

Abstract: Poisson bracket invariants, introduced by Buhovsky, Entov, and Polterovich, are symplectic invariants of quadruples of closed sets whose nonvanishing implies the existence of Hamiltonian trajectories between the sets, with an explicit time-length bound. In this talk, we establish lower bounds on these invariants for certain configurations in completions of Liouville manifolds, expressed in terms of the barcode of wrapped Floer homology. This is inspired by a work of Entov and Polterovich, who obtained analogous results for Lagrangian cobordisms between Legendrians using persistence in Legendrian contact homology. Our main examples concern cotangent bundles, where the quadruple consists of two cosphere bundles of different radii and two cotangent fibers over distinct points, yielding concrete bounds on Hamiltonian trajectories related to perturbed geodesic flows.

Mark Berezovik (Tel Aviv University)

Abstract: In this talk I will discuss a link between billiards in convex planar domains and Hofer’s geometry. For smooth strictly convex billiard tables the Hofer distance between the corresponding billiard ball maps admits an upper bound in terms of a simple geometric distance between the tables. Using this result one can embed the billiard ball map of a convex polygon in the completion, with respect to Hofer’s metric, of the group of smooth area-preserving maps of the annulus. This talk is based on joint work with Konstantin Kliakhandler, Yaron Ostrover, and Leonid Polterovich.