# Workshop on statistical properties of chaotic dynamics in and out of equilibrium

### Description

The workshop aims to bring together experts working on the non-equilibrium properties and related phenomena arising in chaotic dynamical systems. This includes, among others, statistical properties, systems with multiple time scales, thermodynamic formalism, open systems, non-stationary dynamics, random dynamics, and other aspects of transient behaviour.

**SCHEDULE****Monday**

9:00-9:30 : registration

9:30-10:15 : Carlangelo Liverani: Heat equation from a deterministic dynamics

10:20-10:50 : Coffee Break

10:50-11:35 : José Ferreira Alves: Impulsive Lorenz semiflows: physical measures, statistical stability and entropy stability

11:40-12:25 :Dalia Terhesiu: Generalized law of iterated logarithm for dependent processes with superdiffusive behaviour

13:00: Lunch time

15:15-16:00 : Hong-Kun Zhang: Separable Graph Hamiltonian Network

16:00-16:30 : Coffee Break

16:30-17:15 : Jonathan DeWitt: Expanding on average random dynamics on surfaces**Tuesday**

9:30-10:15 : Benoît Saussol: Keplerian Shear in Random Chacon Transformations

10:20-10:50 : Coffee Break

10:50-11:35 : Alexey Korepanov: Intermittent maps: mixing at the boundary between finite and infinite measures.

11:40-12:25 : Marco Lenci: Exactness and K-mixing for extensions of dynamical systems, with application to periodic Lorentz gases

13:00: Lunch time

Free afternoon**Wednesday**

9:30-10:15 : Cecilia Gonzales Tokman: Universal gap growth for Lyapunov exponents of perturbed matrix products

10:20-10:50 : Coffee Break

10:50-11:35 : Péter Koltai: Takens meets Koopman: Linear least squares prediction of nonlinear time series

11:40-12:25 : Tamás Tél: Chaos in dynamical systems subjected to non-negligible parameter drift

13:00: Lunch time

15:15-16:00 : Tanja Schindler: Almost sure limit theorems with applications to non-regular continued fraction algorithms

16:00-16:30 : Coffee Break

16:30-17:15 : Matthew Nicol: Birkhoff sum convergence of heavy-tailed observables to stable laws on Gibbs-Markov systems and applications.

17:20-18:05 : Mark Demers: Equilibrium states and measure of maximal entropy for finite horizon Sinai billiard flows.

18:30- Wine & Cheese reception**Thursday**

9:30-10:15 : Matteo Tanzi: Selfconsistent Transfer Operators: Existence and stability of fixed points in presence of strong coupling

10:20-10:50 : Coffee Break

10:50-11:35 : Gary Froyland: Compound Poisson statistics for dynamical systems via spectral perturbation

11:40-12:25 : Carl Dettmann: Kronecker sequences with many distances

13:00: Lunch time

15:15-16:00 : Yuri Kifer: Limit theorems for signatures

16:00-18:05 : Coffee and Poster session**Friday**

9:30-10:15 : Mike Todd: Multivariate extremes in dynamics

10:20-10:50 : Coffee Break

10:50-11:35 : Wael Bahsoun: Mean field coupled dynamical systems: bifurcations and phase transitions

11:40-12:25 : Ian Melbourne: Natural measures and statistical properties of maps with several neutral fixed points and no physical measure

13:00: Lunch time

Free afternoon

**REGISTRATION**

__Link to the registration page:__

Registration is open **until 10 August 2024**. After registration, we will notify you within two weeks whether your application has been accepted.

The sooner you register, the better. The workshop takes place during the top touristic season in Budapest, so accommodation prices can be expected to increase dramatically with time.

**ACCOMMODATION**

**!!NEW!!** Limited financial support for accommodation is available for students and postdocs. Accommodation is provided in 2-bed rooms. Applications should be submitted as soon as possible but not later than March 31, 2024. Applications should be sent by email to **hdserdos@renyi.hu** including CV and list of publications. Please also indicate your gender, which we need to know because of the 2-bed rooms. Applicants should also arrange for a short letter of recommendation to be sent to the same email address by a faculty member, preferably the advisor/mentor. Please note that those who are supported are expected to attend the whole event. Accommodation is normally supported from Sunday to Saturday at most.

**IMPORTANT!** If you have already applied for accommodation support for the duration of the connecting "School on probabilistic aspects of hyperbolic dynamical systems"

(12 August 2024 - 16 August 2024), and you would like to extend this to the duration of the workshop, just write a short message - there is no need to resend the application material. However, decision about the extension will still be made case by case, limited by the budget.

If you don't need financial support, but you would like our administration to help you with booking (at possibly discounted prices), please write to Mrs. Ildikó Bogáti: **bogati@renyi.hu**

**POSTER SESSION**

There will be a poster session where participants of the workshop can present their work.

If you wish to contribute a poster, please send a prospective title and abstract by email to **hdserdos@renyi.hu**

no later than May 20, 2024. Decision about the accepted contributions will be made by June 15, 2024. We will notify you by email.

**TITLES AND ABSTRACTS**__* José Ferreira Alves: Impulsive Lorenz semiflows: physical measures, statistical stability and entropy stability__

We study semiflows generated via impulsive perturbations of Lorenz flows. We prove that such semiflows admit a finite number of physical measures. Moreover, if the impulsive perturbation is small enough, we show that the physical measures of the impulsive perturbations are close, in the weak* topology, to the unique physical measure of the Lorenz flow. A similar conclusion holds for the entropies associated with the physical measures. Joint work with Wael Bahsoun.__* Wael Bahsoun : Mean field coupled dynamical systems: bifurcations and phase transitions__

I will present recent work, joint with C. Liverani, where we develop abstract framework to study invariant measures of infinite dimensional coupled systems where the coupling strength is not necessarily weak. I will finish the talk with an application to Anosov systems where we observe a phase transition.__* Mark Demers : Equilibrium states and measure of maximal entropy for finite horizon Sinai billiard flows.__

We construct unique equilibrium states for a family of Holder continuous potentials for finite horizon Sinai billiard maps and then lift them to equilibrium states for the corresponding flows. This family includes the potentials corresponding to the measures of maximal entropy for both the billiard map and the billiard flow under a mild condition on recurrence to singularities. This condition is conjectured to hold for generic finite horizon Sinai billiard tables. The MMEs we construct are positive on all open sets and Bernoulli (which implies ergodic and mixing). This is joint work with Viviane Baladi and Jerome Carrand.__* Carl Dettmann : Kronecker sequences with many distances__

Consider the Kronecker sequence {n alpha} where alpha is in the d-torus and 1<=n<=N. The number of distances is #{min_i d(i alpha,j alpha)} for 1<=i,j<=N where d is toral distance using the p-norm for p in [1,infinity]. For d=1 there are at most three distances, which follows from the well known three gap theorem. I will discuss existing and improved bounds for the maximum number of distances in higher dimensions, including both traditional and computer-assisted proofs.__* Jonathan DeWitt : Expanding on average random dynamics on surfaces__

We consider exponential mixing for volume preserving random dynamical systems on surfaces. Suppose that (f_1,...,f_m) is a tuple of volume preserving diffeomorphisms of a closed surface M. We now consider the uniform Bernoulli random dynamical system that this tuple generates on M. We assume that this tuple satisfies a condition called being "expanding on average," which means that there exist C,N>0 such that E[\ln Df^Nv]>C for all unit tangent vectors v. From this assumption we show quenched exponential equidistribution as well as quenched exponential mixing. (This is joint work with Dmitry Dolgopyat)__* Gary Froyland : Compound Poisson statistics for dynamical systems via spectral perturbation__

Abstract: We consider random transformations on a complete metrizable space. The randomness comes from an invertible ergodic driving map acting on a probability space. For a family of shrinking random target sets we consider quenched compound Poisson statistics of returns of random orbits to these random targets. We develop a spectral approach: associated with the random map cocycle is a transfer operator cocycle.

We construct a perturbed cocycle and an associated random variable to count the number of visits to random targets in an orbit. Under suitable assumptions, we show that in the long-orbit limit, the random variables converge in distribution to a compound Poisson distributed random variable. We provide several explicit examples for piecewise monotone interval maps in both the deterministic and random settings.__* Cecilia Gonzalez Tokman : Universal gap growth for Lyapunov exponents of perturbed matrix products__

We present recent results on quantitative simplicity of the Lyapunov spectrum of d-dimensional bounded matrix cocycles subjected to additive random perturbations. In dimensions 2 and 3, we establish explicit lower bounds on the gaps between consecutive Lyapunov exponents of the perturbed cocycle, depending only on the scale of the perturbation. In arbitrary dimensions, we show existence of a universal lower bound on these gaps. A novelty of this work is that the bounds provided are uniform over all choices of the original sequence of matrices, making no stationarity assumptions. Hence, our results apply to random and sequential dynamical systems alike. Joint work with Jason Atnip, Gary Froyland and Anthony Quas.__* Yuri Kifer : Limit theorems for signatures__

I will discuss moment and almost sure strong invariance principles and laws of iterated logarithm for normalized multiple iterated sums and integrals involving centered stationary vector processes with some weak dependence properties which can be generated, in particular, by dynamical systems. The results are applicable, in particular, to hyperbolic and expanding dynamical systems with their Gibbs measures, Gibbs-Markov maps and some systems which can be represented via the Young towers construction. Sequences of iterated sums and integrals were called signatures in recent papers in rough paths, data science and machine learning.__* Péter Koltai: Takens meets Koopman: Linear least squares prediction of nonlinear time series__

The least squares linear filter, also called the Wiener filter, is a popular tool to predict the next element(s) of time series by linear combination of time-delayed observations. We consider observation sequences of deterministic dynamics, and ask: Which pairs of observation function and dynamics are predictable? If one allows for nonlinear mappings of time-delayed observations, then Takens' well-known theorem implies that a set of pairs, large in a specific topological sense, exists for which an exact prediction is possible. We show that a similar statement applies for the linear least squares filter in the infinite-delay limit, by considering the forecast problem in an invertible ergodic-theoretic setting and the Koopman operator on square-integrable functions.__* Alexey Korepanov: Intermittent maps: mixing at the boundary between finite and infinite measures.__

Intermittent maps admit a unique absolutely continuous invariant measure which depending on the parameter can be finite or infinite. In both cases, mixing properties such as decay of correlations have been thoroughly investigated by many. I'll talk about mixing rates and proofs that survive the transition between finite and infinite measure regimes.__* Marco Lenci : Exactness and K-mixing for extensions of dynamical systems, with application to periodic Lorentz gases__

Exactness and the K-property (a.k.a. K-mixing) are strong ergodic properties whose significance is that dynamical system progressively loses all information about its initial conditions. These properties were the subject of intense investigation in the second half of last century before being somewhat superseded by the stronger Bernoulli property, which was at the time proved for a number of popular systems, including many billiards. But the Bernoulli property can only be formulated for dynamical systems preserving a finite measure, while exactness and the K-property work equally well in infinite-measure contexts, which partly explains a revived interest in them.

I will present some general theorems on the exact or K decomposition of extensions of exact or K dynamical systems. One immediate application is that, in large generality, a recurrent periodic Lorentz gas whose finite-measure factor is proved to be hyperbolic is K-mixing. This includes virtually all previously studied 2-dimensional periodic Lorentz gases and d-dimensional periodic Lorentz tubes, as well as many d-dimensional periodic Lorentz slabs.

Joint work with Daniele Galli.__* Carlangelo Liverani: Heat equation from a deterministic dynamics__

I’ll describe a derivation of the heat equation in the thermodynamics limit, with a diffusive scaling, from purely deterministic dynamics satisfying Newton's equations under a time-dependent external field.

(Work in collaboration with G.Canestrari and S. Olla)__* Ian Melbourne: Natural measures and statistical properties of maps with several neutral fixed points and no physical measure__

We show that infinite measure-preserving intermittent maps that do not admit physical measures nevertheless exhibit strong statistical properties. In particular, we prove existence of a distinguished "natural" measure such that the pushforwards of any absolutely continuous probability measure converge to it. We also obtain results on almost sure and distributional convergence for empirical measures.

(Joint with Douglas Coates and Amin Talebi.)__* Matthew Nicol : Birkhoff sum convergence of heavy-tailed observables to stable laws on Gibbs-Markov systems and applications.__

We use a Poisson point process approach to prove distributional convergence to a stable law for heavy-tailed observables on Gibbs-Markov maps. A key result is to verify a standard mixing condition, which ensures that large values of the observable dominate the time-series. Stable limit laws for observables on dynamical systems have been established in two settings: ``good observables'' on slowly mixing non-uniformly hyperbolic systems and ``bad'' observables (unbounded with fat tails) on fast mixing dynamical systems.

As an application we investigate the interplay between these two effects in a class of intermittent-type maps.__* Benoît Saussol: Keplerian Shear in Random Chacon Transformations__

Keplerian shear was recently introduced by Damien Thomine. Loosely speaking, this property can be thought of as a form of mixing for non-ergodic systems. In this study, we examine a parameterized family of Chacon transformations and demonstrate that they satisfy the Keplerian shear property. Each transformation in the family is not mixing but only weakly mixing. Specifically, there are rigidity times during which mixing cannot occur. The fundamental idea behind this work is that these rigidity times are highly dependent on the parameter, leading to a collective behavior where mixing predominates. Surprisingly, the proof utilizes a non-stationary version of the local limit theorem. This is a joint work with Arthur Boos (I2M, Marseille).__* Tanja Schindler : Almost sure limit theorems with applications to non-regular continued fraction algorithms__

We consider an ergodic measure preserving dynamical system (X,T) and an observable f mapping X to the reals. By a theorem by Aaronson there is no strong law of large numbers if either 1) X is a probability measure space and f is non-integrable or if 2) X has infinite measure und f is integrable. While in the situation 1) one can use trimming, i.e. deleting a number of the largest entries, to still obtain a strong law of large numbers, in case of 2) it is possible to add additional summands to obtain a strong law of large numbers. In this talk we will study the situation 2) and also the situation of an infinite measure space and an observable which is non-integrable on a finite measure set.

Examples of such a situation are different non-standard continued fraction digits like backwards continued fractions. This is joint work with Claudio Bonanno.__* Matteo Tanzi : Selfconsistent Transfer Operators: Existence and stability of fixed points in presence of strong coupling__

Selfconsistent transfer operators (STOs) are nonlinear operators describing the thermodynamic limit of mean-field coupled maps, and are the discrete time analogues of McKean-Vlasov equations. They allow to model and study the emergent behaviour of interacting chaotic units in discrete time and, more generally, of systems whose evolution is not Hamiltonian or for which there are no evident conserved quantities.

The study of the dynamics of STOs gives information on the emergent behaviour in the thermodynamic limit and, in particular, on the existence and stability of equilibrium measures. Studies of STOs have mostly been limited to the case of small coupling where the STO is close to a linear operator. In this talk, I will describe recent results providing sufficient conditions for the existence and stability of fixed points in the strong coupling regime where the STO presents “genuinely” nonlinear dynamics with bounded basins of attraction and multiple fixed points.__* Tamás Tél : Chaos in dynamical systems subjected to non-negligible parameter drift__

Chaotic systems subjected to parameter drift of non-adiabatic rate are an area where many methods of traditional chaos theory are not applicable. Periodic orbit expansion cannot be applied since no periodic orbits exist, nor do long-time limits, since they might lead out of the range of validity of the model. Such systems are non-ergodic, time and ensemble averages are different, the ensemble approach is superior to the single-trajectory view. In dissipative systems, attractors are time-dependent in a non-periodic fashion. In the phase space, the analysis can be based on stable and unstable foliations, their intersections defining a Smale horseshoe. Because of the drift, foliations are also time-dependent. We give a description of the time-dependent natural measure, illustrated by numerical examples. As a quantitative measure for the strength of chaos, the so-called ensemble-averaged pairwise distance (EAPD) can be evaluated at any time instant. The slope of this function can be considered the instantaneous (largest) Lyapunov exponent. We find that the time-dependent attractor coincides with the unstable foliation, and can consist of more than one component: a chaotic one, a time-dependent chaotic saddle, and moving regular attractor points. The basin boundaries of such points contain the time-dependent stable foliation.__* Dalia Terhesiu: Generalized law of iterated logarithm for dependent processes with superdiffusive behaviour__

I will present recent results which naturally generalize the strong invariance principle and the law of iterated logarithm to superdiffusive processes with heavy dependencies, focusing on the set-up of the Lorentz gas with infinite horizon. This is joint work with Péter Bálint.__* Mike Todd : Multivariate extremes in dynamics__

A sequence of random variables in R^d can have extreme values in each of the d coordinates. In interesting cases, these extremes can have clusters of extremes and these have been studied when the random variables come from dynamical systems. However, until now the issue of how an extreme value in one (or more) of the d coordinates may trigger an extreme value in another coordinate, possibly after some time delay, had not been addressed. This natural question is answered in work with Aimino, Freitas and Freitas where we find a generalised extremal index function with an appealingly canonical form. I will outline our theory and show how it applies to some elementary dynamical systems models.__* Hong-Kun Zhang (University of Massachusetts Amherst /Great Bay University): Separable Graph Hamiltonian Network__

In this talk, we introduce a deep learning approach leveraging graph neural networks to address the challenges posed by nonlinear lattice models, which are fundamental in studying complex dynamical systems. Our method represents lattice systems as graphs, capturing intricate nonlinear relationships that traditional techniques often miss. By incorporating separable Hamiltonians, our model excels in trajectory prediction and energy conservation, essential for accurately modeling dynamical behavior. Through extensive testing on classic nonlinear oscillator lattice problems, including the Fermi-Pasta-Ulam system, our approach has successfully captured interaction forces.