Author: Daniel Tsodikovich

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This difficulty arises because standard transversality methods are no longer applicable [4], raising the conjecture of whether classical results on generic non-integrability extend to the setting of potential fields.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

Santiago Barbieri (University of Girona)

The steepness property is a local geometric transversality condition on the gradient of a $C^2$-function which proves fundamental in order to ensure the stability of sufficiently-regular nearly-integrable Hamiltonian systems over long timespans. Steep functions were originally introduced by Nekhoroshev, who also proved their genericity. Namely, given a pair of positive integers $r,n$, with $r$ high enough, and a point $I_0\in \R^n$, the Taylor polynomials of those $C^{2r-1}$ functions which are not steep around $I_0$ are contained in a semi-algebraic set of positive codimension in the space of polynomials of $n$ variables and degree bounded by $r$. The demonstration of this result was originally published in 1973 and has been hardly studied ever since, probably due to the fact that it involves no arguments of dynamical systems: it makes use of quantitative reasonings of real-algebraic geometry and complex analysis.  In the first part of the presentation, I will explain the proof of the genericity of steepness by making use of modern tools of real-algebraic geometry: this allows to clarify the original reasonings, that were obscure or sketchy in many parts. In particular, Yomdin’s Lemma on the analytic reparametrization of semi-algebraic sets, together with non trivial estimates on the codimension of certain algebraic varieties, turns out to be the fundamental ingredient to prove the genericity of steepness. The second part of the presentation will be devoted to the formulation of new explicit algebraic criteria to check steepness of any given sufficiently regular function, which constitutes a very important result for applications, as the original definition of steepness is not constructive. These criteria involve both the derivatives of the studied function up to any given order and external real parameters that, generically, belong to compact sets. 
References: 
S. Barbieri “Semi-algebraic geometry and generic Hamiltonian stability”, Adv. Math. 482 C, 2025N. N. Nekhoroshev. Stable lower estimates for smooth mappings and for gradients of smooth functions. Math USSR Sb., 19(3):425–467, 1973.

Mathieu Helfter (ISTA)

In this talk, I will present a framework for multifractal analysis of infinite-dimensional measures. This us based on the notion of “scales,” which are finite metric invariants that generalize classical notions such as Hausdorff dimension and pointwise dimension of measures to both finite and infinite-dimensional spaces. We will focus in particular on the paradigmatic example of Brownian motion, whose multifractal spectrum can be described using measures associated with fractional Brownian motions. Since the Wiener measure cannot serve as a Haar measure, it is necessarily concentrated on a rich set of non-typical trajectories, whose scaling irregularities reveal its multifractal nature. We will also discuss several new questions and problems arising in this framework. The talk is based on joint work in progress with Aihua Fan.

Abel Komalovics (University of Budapest)

If we have a family of expanding maps with unique ACIMs, then the ACIM stability of the family means that if the maps are converging to a limit map, then the densities converge to the density of the limit map. There are multiple papers showcasing ACIM instability, but all of them use methods specific to the families introduced in the papers. We develop a unified framework that allows us to use the tools of analysis of metastable maps. This way we can calculate the limit density, the finite dimensional distribution of the wandering of points and the diffusion coefficient.

Nicanor Carrac (Jagiellonian University)

This talk is about topological $[T,T^{-1]$ systems, which are skew products built by taking a subshift in the base, a continuous cocycle to the integers, and an arbitrary invertible system in the fiber. The main result states that under suitable conditions the entropy of the fiber system can be recovered as the slow entropy of the skew product with a well-chosen scale. This is a topological analogous of existing results for measure-preserving   $[T,T^{-1]$ systems (Heicklen, Hoffman, & Rudolph; Ball; Austin). The main novelty it can be applied with zero-entropy systems in the base. This is based on the preprint https://arxiv.org/abs/2506.17932

Andreas Vollmer (Hamburg Universität)

Abstract: We introduce a geometric framework for second-order (maximally) superintegrable Hamiltonian systems on Riemannian manifolds of any dimension, encoding the system in a symmetric cubic tensor field (work with J. Kress and K. Schöbel). This approach allows one to interpret Stäckel transformations (coupling constant metamorphosis) of superintegrable systems in terms of a conformal geometry (or, more precisely, a Weylian structure).
The general framework will be exemplified for systems with a maximal number of compatible potentials and of linearly independent integrals of motion (“abundant systems”). This class includes the Smorodinski-Winternitz system, for instance. We find a natural correspondence between abundant systems and affine hypersurfaces (work with V. Cortés). On spaces of constant sectional curvature, abundant systems have an underlying Hessian structure (with J. Armstrong), and on flat space they correspond to solutions of the Witten-Dijkgraaf-Verlinde-Verlinde equation, allowing their identification with Manin-Frobenius manifolds.

Piotr Oprocha (AGH Krakow)

Abstract: A very useful technique called BBM (Brown-Barge-Martin), incorporates inverse limits and natural extensions of the underlying bonding maps to embed attractors in manifolds.
The original idea goes back to the paper of Barge and Martin, where the
authors constructed strange attractors from a wide class of inverse limits.
One of the crucial steps for this technique to work is the usage of Brown’s approximation theorem.
Recently, this technique was extended to produce a parameterized family
of strange attractors.
In this talk we will present a few possible applications of BBM technique in a construction of concrete examples.

Michael Entov (Technion – Israel Institue of Technology)

Abstract: I will discuss how a filtered version of the Legendrian contact homology (a major object of study in contact topology) can be used to obtain new information about contact Hamiltonian flows on contact manifolds. The applications concern conformal factors of contactomorphisms and trajectories of contact Hamiltonian flows connecting two disjoint Legendrian submanifolds or starting on one of them and asymptotic to the other.

This is a joint work with L.Polterovich.

Eran Igra, Valerii Sopin (SIMIS – Shanghai Institute for Mathematics and Interdisciplinary Sciences)

Consider a rectangle map $H:ABCD\to\mathbb{R}^2}$ s.t. $H$ is a Smale Horseshoe map. Now, consider another map $G:ABCD\to\mathbb{R}^2$ s.t. $G$ is the identity (or more generally, some other homeomorphism with “simple” dynamics). By a classical result due to James Yorke and Kathleen Alligood we know that as we deform $G$ isotopically to $H$, more and more periodic orbits appear via period-doubling cascades, culminating in the chaotic dynamics of the Smale Horseshoe – i.e., we have a period-doubling route to chaos. That being said, due to the two-dimensional structure of the phase space we are free to choose many isotopies deforming $G$ to $H$ – and all feature a period-doubling route to chaos. This leads us to ask the following, natural question: can we somehow differentiate between different two-dimensional period-doubling routes to chaos?

It is precisely this question that we study in this talk. Using a mixture of Representation-Theoretic, Complex-Analytic, and Braid Theoretical tools we define topological invariants that describe how a given isotopy progresses from order into chaos. Following that, we exemplify how these invariants can be used to analyze the dynamical complexity of certain examples, including the Henon map – if time permits, we also discuss how these ideas can be generalized to a wider context.

Based on work in progress.

Dongchen Li (Imperial College London)

A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some central subspace has a non-empty interior and thus has a higher topological dimension than the set itself.We show that, for any symplectic C^r-diffeomorphism (where r is sufficiently large and finite, or r=∞,ω) of a 2N-dimensional (N>1) symplectic manifold, symplectic blenders can be obtained by an arbitrarily small symplectic perturbation near any one-dimensional whiskered KAM-torus that has a homoclinic orbit. Using this result, we prove that non-transverse homoclinic intersections between invariant manifolds of a saddle-center periodic point (i.e., it has exactly one pair of complex multipliers on the unit circle) are persistent in the following sense: the original map is in the C^r closure of a C¹ open set in the space of symplectic C^r-diffeomorphisms, where maps having such saddle-center homoclinic intersections are dense. These results also hold for Hamiltonian flows in the corresponding settings.