Wild dynamics on manifold

• Pierre Berger (Université Sorbonne, France)

Room: Aula Dini We survey a few results on differentiable, symplectic, or analytic wild dynamics.

When Contraction Is Not Enough: Symbolics, endpoints and (in)stability of local IFSs

• Paulo Varandas (Universidade de Aveiro, Portugal)

Room: Aula Dini Classical contractive iterated function (IFSs) systems enjoy a remarkably rigid picture: a unique attractor, full symbolic coding, and robust asymptotic behavior. This picture changes substantially when the maps are defined only on prescribed local domains. In this talk, we present a framework for contractive local iterated function systems and investigate the structure and stability of their attractors. In contrast with the classical setting, the local attractor may contain endpoints corresponding to finite admissible orbits, its symbolic dynamics need not be of finite type, and different dynamical regimes may coexist within the same attractor, leading to a new notion of combinatorial instability. Joint work with E. Oliveira, UFRGS-Brazil.

Ring's dynamics: from KAM to bifurcation theory

• Alessandra Celletti (Università di Roma Tor Vergata)

Room: Aula Dini This talk presents a Hamiltonian framework for modelling the dynamics of ring particles orbiting celestial bodies, encompassing both nearly spherical planets and markedly non-spherical asteroids. The Hamiltonian is formulated using the so-called epicyclic variables. The subsequent analysis focuses on the dynamics in the vicinity of resonances, specifically on commensurabilities between the orbital frequency of the ring particles and the rotational frequency of the central body. The analysis employs an integrated framework combining normal form techniques, multi-scale Kolmogorov–Arnold–Moser (KAM) theory, and bifurcation theory to investigate the stability properties of resonant structures across a range of dynamical regimes. This talk refers to joint works with Irene De Blasi and Sara Di Ruzza.

Synchronization in transformers

• Marco Romito (Università di Pisa)

Room: Aula Dini The first part of the talk will introduce a mathematical framework for transformers architectures, focusing in particular on how information travels through the layers of the network. In the second part we will briefly discuss infinite-width, infinite-depth and infinite-tokens limits. Finally, the third part will focus on synchronization and some noise-induced phenomena.

The Thurston algorithm for real entire transcendental post-singularly finite maps

• Sebastian van Strien (Imperial College London, UK)

Room: Aula Dini Thurston’s characterisation theorem gives a necessary and sufficient condition for when a branched covering map of the sphere (for which the orbits of the branch points have finite cardinality) can be realised by a rational map. In spite of progress, an analogous result for entire maps of the complex plane seem to be not yet known. In this talk, I will discuss a different approach in the setting of real entire maps whose post-singular set is real.

Trimmed sums, extravagances and extreme historic behaviour

• Tanja I. Schindler (University of Exeter, UK)

Room: Aula Dini We consider an irrational rotation R_α on the torus (R_α(x)= x+α mod 1) and the non-integrable observable f(x)=1/x-1/(1-x) and study the Birkhoff sum S_n=f+...+f(R_α^(n-1)). As by Aaronson's theorem it is not possible to obtain strong convergence for S_n/d_n where d_n is any norming sequence, we look at the trimmed sum where we delete one or more of the maximal terms to obtain strong convergence. Depending on the Diophantine properties of α we may need to delete only one or many of the maximal entires of the Birkhoff sum to obtain strong convergence. Moreover, we look at extravagance, i.e. the limsup behaviour of f(R_α^n) / S_n and show under which conditions it equals ∞ or 0. These results have consequences in studying a reparametrization (T_t) of the linear flow (L_t) with direction (1,α) on the two torus T2 with function φ, where φ is a smooth non-negative function that has exactly two (non-degenerate) zeros at p and q. We prove that for a full measure set (α,p,q)∈ T×T^2×T^2 the special flow (T_t) exhibits extreme historic behavior proving a conjecture given by Andersson and Guihéneuf. The talk is based on joint work with Max Auer and with Adam Kanigowski.

Partially hyperbolic dynamics in the 3-body problem

• Marcel Guardia (Universitat de Barcelona, Spain)

Room: Aula Dini In this talk I will explain how to construct symplectic blenders for two classical Hamiltonian systems: the 3-body problem and its restricted version. These objects lead to a strong form of topological instability. This is a joint work with Jaime Paradela.

Classification of Henon maps with strange attractors via a stable manifold

• Sonia Stimac (University of Zagreb, Croatia)

Room: Aula Dini In an earlier work with Boronski, we classified (up to conjugacy) the Henon maps with strange attractors in terms of three invariants that we introduced for them: (a) kneading sequences, (b) pruned trees, and (c) folding patterns of the unstable manifold of the hyperbolic fixed point $X$ in the attractor. In my talk, I will introduce yet another way to determine conjugacy classes of these maps, this time via the stable manifold $W^s$ of $X$. We consider a region of dissipation $D$ for the Henon map and study the connected components of $D \cap W^s$. To each such component, we assign a separation type and prove that two Henon maps are conjugate if and only if their corresponding components share the same separation type.       This is joint work with Jan Boronski.

Wide Deep Neural Networks as Random Dynamical Systems: From Gaussian Limits to Least Action

• Dario Trevisan (Università di Pisa)

Room: Aula Dini Wide fully connected neural networks admit a remarkably tractable probabilistic description at large width. At fixed depth, their outputs converge to Gaussian processes, while the empirical covariance kernels of successive layers evolve according to a deterministic nonlinear recursion. Interpreting depth as a discrete time variable, this places the standard neural-network Gaussian process (NNGP) theory naturally in the language of dynamical systems. I will discuss how large-deviation theory enriches this picture. Rather than describing only the typical kernel trajectory, a large-deviation principle assigns a cost to atypical transitions between successive layer kernels. This leads to a variational formulation on paths: the usual NNGP dynamics appears as the zero-cost trajectory, while atypical outputs and representations are obtained by minimizing an action subject to suitable endpoint constraints. In this sense, large deviations provide a least-action counterpart to the forward NNGP recursion. This perspective suggests connections with stability and criticality of the kernel dynamics, including the usual edge-of-chaos picture, as well as with Bellman principles, optimal trajectories and rare transitions in deep networks. I will illustrate these ideas with examples for fully connected networks and discuss how conditioning or statistical losses naturally enter the same framework as additional variational costs. The talk is partly based on joint works with Claudio Macci, Barbara Pacchiarotti, Katerina Papagiannouli, Giovanni Luca Torrisi and Giuseppe Pio Zito.

Identifying Almost-Invariant Regions in Space Debris Dynamics via Transfer Operators

• Riccardo De Pascalis (Università del Salento)

Room: Aula Dini In recent years, techniques from ergodic theory have been successfully applied to investigate a wide range of real-world phenomena. In this work, we explore how the spectral properties of transfer operators can provide insight into transport in space debris dynamics, with the aim of identifying regions that resist mixing over long timescales. We consider an ensemble of small orbiting debris fragments and analyse their dynamics using a spherical Poincaré section centred on the Earth. The associated transfer operator is approximated by a finite-dimensional Markov operator via Ulam's method. We investigate its spectral properties and identify almost-invariant sets. The theoretical framework is applied to a set of space debris in the lower region of medium Earth orbit (MEO) subject to the Earth’s gravitational field and its oblateness. This is a joint work with Anna Maria Cherubini (Università del Salento) and Gary Froyland (University of New South Wales).

Nekhoroshev Theorem for time quasiperiodic perturbations of integrable systems

• Beatrice Langella (Università di Pisa)

Room: Aula Dini Consider a small amplitude perturbation of an analytic integrable Hamiltonian system: Nekhoroshev theorem ensures that, if the unperturbed part of the Hamiltonian fulfills a suitable non-degeneracy assumption, known in literature as steepness, all the actions are quasi-preserved for time scales that are exponentially long in the perturbative parameter. What happens when the perturbative term in the Hamiltonian is also allowed to depend on time ? We study the case of time quasi-periodic perturbations and we prove that a Nekhoroshev type result still holds, under the assumption that the system fulfills a different non-degeneracy condition, originally introduced by Nekhoroshev and called P-steepness. Our approach is based on an extension to the time dependent case of the construction exhibited by Guzzo, Chierchia and Benettin in their proof of classical Nekhoroshev theorem. One of the main difficulties is to deal with the more complex geometry of resonances of the time dependent case. This talk is based on a joint work with D. Bambusi, S. Barbieri, and M. Giralt.

Geometric Properties of Higher Dimensional Solenoidal Attractors

• Daniele Galli (University of Zurich, Switzerland)

Room: Aula Dini We study the skew product systems $T:\mathbb{S}^1\times\mathbb{R}^d\to\mathbb{S}^1\times\mathbb{R}^d$, $T(x,y)=(\ell x, A y+\phi(x)),$ where $\ell\geq 2,$ $A \in GL_d(\mathbb{R})$, with $\rho(A)<1$, and $\phi \in C^r(\mathbb{S}^1, \mathbb{R}^d)$. We show that for a generic $\phi$: if $|\det(A)|\ell<1$, the Hausdorff dimension of the solenoid attractor and the SRB measure dimension equals the affinity dimension; if $|\det(A)|\ell>1$, the SRB measure is absolutely continuous. In the latter case we investigate the Sobolev regularity of the density of the SRB measure. We introduce a derivative dispersion condition, which replaces the usual transversality condition. This is a joint work with E.Passaglia and A.Ulliana

Nonautonomous differential equations in the presence of bounded noise

• Iacopo Longo (Imperial College London, UK)

Room: Aula Dini Nonautonomous systems subject to noise arise naturally in many applied contexts. Particularly relevant are those featuring time-dependent parameters and uncertainties, which may drive the emergence of tipping points. In this talk, we focus on the case of bounded noise, where the dynamics can be captured topologically via deterministic set-valued dynamical systems: initial conditions are evolved under all admissible noise realisations, abstracting from probabilistic details. However, as these systems operate on the space of nonempty compact subsets—a space lacking Banach structure—they pose substantial analytical and numerical challenges. In particular, bifurcation analysis of attractors remains a significant obstacle. To address this, we derive a higher-dimensional single-valued deterministic dynamical system which characterises the evolution of the boundary of invariant sets of certain differential inclusions.

Building chaos in billiards with central forces

• Irene De Blasi (Università di Torino)

Room: Aula Dini In recent years, a large community of mathematicians has become increasingly interested in the study of billiards with potential, investigating how the transition from straight geodesics to curved ones can influence the overall dynamics. In particular, the presence of singularities can greatly enrich the dynamics, leading to chaotic phenomena not observed in the case of classical billiards. In this talk, we propose the construction of chaotic dynamics in billiard systems with a central potential, demonstrating how, in various cases, the existence of a symbolic dynamics (and thus of positive topological entropy) can be proven. Works in collaboration with S. Baranzini, V. Barutello, A.M. Cherubini, A. Takeuchi and S. Terracini.

A rigidity result for Kepler billiards

• Vivina Barutello (Università di Torino)

Room: Aula Dini We study the integrability of Kepler billiards, mechanical systems in which a particle moves under the influence of a Keplerian potential and undergoes elastic reflections at the boundary of a strictly convex planar domain with smooth boundary. We prove that, with the possible exception of a single position of the gravitational center, analytic integrability at any energy regime can occur only when the domain is an ellipse and the center is located at one of its foci. This result provides a partial affirmative answer to the Keplerian counterpart of the classical Birkhoff–Poritsky Conjecture. This research is in collaboration with S. Baranzini, I. De Blasi and S. Terracini.

Linear response, dicontinuities and holes

• Giovanni Canestrari (University of Toronto, Canada)

Room: Aula Dini Linear response means that the equilibrium measures of chaotic dynamical systems react to perturbations linearly in the perturbation strength. After reviewing some obstructions to linear response for 1D expanding maps, I will present some results on linear response for discontinuous perturbations and systems with holes in higher dimensions.

Mean-field derivation of a Vlasov-type equation in a Sinai billiard

• Roberto Castorrini (Università della Tuscia)

Room: Aula Dini We study a system of mean-field coupled particles moving in a Sinai billiard under the influence of an electric field and a Gaussian thermostat, together with the associated nonlinear Vlasov-type equation. We prove existence and uniqueness of the macroscopic evolution through a fixed-point formulation involving self-consistent transfer operators, and show that it arises as the mean-field limit of the microscopic particle system.

Extreme Value Theorem for geodesic flow on the quotient of the theta group

• Seul Bee Lee (Seoul National University, South Korea)

Room: Aula Dini The extreme value theorem (EVT) is a theory concerning the distribution of the maximum of a sequence of random variables. For classical continued fractions, an EVT has been established, describing the limiting distribution of large partial quotients. Their significance is further explained by the correspondence with geodesic flow on the modular surface, where large partial quotients can be seen as excursions to the cusp. In this talk, we first establish an EVT for a modified version of the even continued fraction expansion. This result is then used to derive the EVT for the geodesic flow on the hyperbolic surface associated with the theta group, which has two cusps. This is joint work with Jaelin Kim and Seonhee Lim.

Weighted Products of Digits in Gauss-like Iterated Function Systems

• Ayreena Bakhtawar (Universität Wien)

Room: Aula Dini The growth of continued-fraction digits is closely related to Diophantine approximation. Products of consecutive digits arise, in particular, in questions concerning improvements to Dirichlet’s theorem. These connections motivate the study of the Hausdorff dimension of sets defined by prescribed growth of digit products. In this talk, I will discuss weighted products of digits in d-decaying Gauss-like iterated function systems, a class that includes continued fractions and Lüroth expansions. We determine the Hausdorff dimension of sets on which a finite product of digits, raised to arbitrary positive powers and taken at arbitrary fixed positions, exceeds a prescribed function infinitely often. In particular, our result resolves the weighted dimension problem beyond two consecutive digits in the exponential-growth regime, which remained open even for the classical Gauss and Lüroth maps. A second aspect of the work is that bounded distortion is not assumed. We introduce a weaker property, almost tempered distortion, and use it to prove the existence of geometric pressure defined by cylinder lengths. This is joint work with Michał Rams.

Isoenergetic degeneracy generically creates meandering invariant tori

• Illya Koval (Universität Wien, Austria)

Room: Aula Dini Consider the set of isoenergetically degenerate integrable Hamiltonians with two degrees of freedom. We show that a cusp-generic perturbation of a generic Hamiltonian in this set gives rise to meandering invariant tori - embedded Lagrangian tori which are not graphs. Moreover, an exponentially dense subset of perturbations admits higher order meandering tori, of all orders from two to infinity. These infinite order meanders have an endless nested structure.