Author: Daniel Tsodikovich

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.

Tri Minh Le (TU Wien)

Abstract: Eikonal equations in metric spaces have strong connections with the local slope operator (or the De Giorgi slope). This talk discusses an analogous model associated with the global slope operator, expressed as $\lambda u + G[u] = \ell$, where $\lambda \geq 0$. The case $\lambda=0$ is naturally related to the so–called discrete weak KAM theory. Under mild assumptions on the metric space $X$ and the given function $\ell$, we establish the well–posedness of this equation. Our techniques further imply a new integration formula based on the global slope operator.

Laurent Stolovitch (Université Côte d’Azur)

In this article, we consider perturbations of isometries on a compact Riemannian manifold $M$.
We investigate the smooth (resp. analytic) rigidity phenomenon of groups of these isometries. As a particular case, we prove that if a finite family of smooth (resp. analytic) small enough perturbations is simultaneously conjugate to the family of isometries via a finitely smooth diffeomorphism, then it is simultaneously smoothly (resp. analytically) conjugate to it whenever the family of isometries satisfies a Diophantine condition.
Our results generalize the rigidity theorems of Arnold, Herman, Yoccoz, Moser, etc. about circle diffeomorphisms which are small perturbations of rotations as well as Fisher-Margulis’s theorem on group actions satisfying Kazhdan’s property (T). This a collaboration with Zhiyan Zhao.

Sonja Hochloch (University of Antwerp)

Abstract: In this talk, we will explain how the goal of symplectically classifying integrable systems on 4-dimensional manifolds with underlying S^1-symmetry led to the discovery and subsequent study of a new type of bifurcation, called double flip bifurcation. We give a normal form for systems going through this bifurcation and calculate some examples.

The bifurcation part of this talk is based on the joint work arXiv:2511.12086 with K. Efstathiou (Duke Kunshan University) and Tobias Henriksen (Antwerpen & Groningen).

Silvia Radinger (University of Vienna)

Abstract: In this talk we study measure-theoretical rigidity and partial rigidity for classes of Cantor dynamical systems including Toeplitz systems. The use of Bratteli-Vershik diagrams enables us to control the structure of the ergodic invariant measures.
Among other things, we will analyze different Toeplitz systems for their rigidity, show that there exist uniquely ergodic Toeplitz systems which have zero entropy and are not partially measure theoretically rigid and construct a Toeplitz system which has countably infinitely many ergodic invariant probability measures that are rigid with the same rigidity sequence. Further we show general conditions under which a Bratteli-Vershik system is rigid and give examples of non-rigidity.
This talk is based on joint work with Henk Bruin, Olena Karpel and Piotr Oprocha.

Mihajlo Cekić (Université Paris-Est Créteil)

Abstract: We will discuss the speed of mixing of extensions of chaotic (Anosov) flows to principal bundles. For example, this includes the frame flow on the bundle of orthonormal frames over a negatively curved Riemannian manifold, and our result in this setting guarantees that ergodicity of the frame flow implies its rapid mixing, that is, mixing faster than C_N t^{-N} for any N > 0 (here t > 0 denotes time and C_N > 0). In the second situation of interest, we will consider extensions to Z^d covers; in this case, we will prove the optimal speed of mixing Ct^{-d/2}, and moreover we will provide the full asymptotic expansion in decaying powers of the time t. To prove these results we develop a novel semiclassical calculus on principal G-bundles that we call the Borel-Weil calculus, where the semiclassical parameters correspond to the highest roots parametrizing irreducible representations of G. The calculus can be shown to have further applications to hypoellipticity and quantum ergodicity of horizontal (sub-)Laplacians. Joint works with Thibault Lefeuvre and Sebastián Muñoz Thon.

Manuel Garzón (ICMAT-CSIC)

The question of whether a Hamiltonian system is typically integrable or chaotic is a central topic in dynamical systems, which traces back to the pioneering works of Poincare in Celestial Mechanics. A satisfactory picture of the typical dynamics of such systems did not emerge until the 1970s, when Markus and Meyer [5] established that a generic (in the Baire category sense) Hamiltonian system on a compact symplectic manifold is neither integrable nor ergodic. On the contrary, the case of natural Hamiltonian systems is much less studied, in spite of its central relevance in mathematical physics. Specifically, a natural Hamiltonian corresponds to the situation in which the symplectic manifold is the cotangent bundle of a manifold M, and the Hamiltonian is given by the sum of a fixed kinetic energy term and a potential field V ∈ C∞(M;R). It is known that a generic potential field on a compact manifold is non-ergodic [4]. Moreover, near the potential
maximum, the system may exhibit positive topological entropy under suitable conditions [1, 2]. Nevertheless, the fundamental question of whether motion at low energy levels is typically integrable or chaotic remains open to date.
This difficulty arises because standard transversality methods are no longer applicable [4], raising the conjecture of whether classical results on generic non-integrability extend to the setting of potential fields. In this talk we shall present the main result of [3] which shows that, on each low energy level, the natural
Hamiltonian system defined by a generic smooth potential V on T2 exhibits an arbitrarily high number of hyperbolic periodic orbits and a positive-measure set of invariant tori. To put this result in perspective, let us recall that hyperbolic periodic orbits are the natural starting point to establish the presence of chaos in dynamical systems.

References
[1] S.V. Bolotin, P.H. Rabinowitz, A variational construction of chaotic trajectories for a Hamiltonian system on a
torus, Boll. Uni. Mat. Ital. 1B (1998) 541–570.
[2] S.V. Bolotin, P.H. Rabinowitz, A variational construction of chaotic trajectories for a reversible Hamiltonian
system, J. Diff. Eqs. 148 (1998) 364–387.
[3] A. Enciso, M. Garz´on, D. Peralta-Salas: Low-energy dynamics in generic potential fields: Hyperbolic periodic
orbits and non-ergodicity. Preprint, 2025
[4] L. Markus, K. R. Meyer. Generic Hamiltonian dynamical systems are not nor ergodic. Proc. Conf. on Nonlinear
Oscillation Kiev, p. 311-332, 1969.
[5] L. Markus, K. R. Meyer: Generic Hamiltonian dynamical systems are neither integrable nor ergodic, Mem. Amer.
Math. Soc. 144, 1974