Regular and stochastic behaviour in dynamical systems

12-14 February 2020, University of Rome Tor Vergata

What and why?

This is the kick-off meeting for the PRIN grant "Regular and stochastic behaviour in dynamical systems" . The primary aim is to initiate and support collaboration between the various dynamical systems groups in Italy. We'll have some seminars but, most importantly, plenty of spare time for informal discussions.

Location: Department of Mathematics, University of Rome Tor Vergata. There are various options to arrive at the department by public transport, some more info is here.
Organizers: Carlangelo Liverani, Alfonso Sorrentino and Oliver Butterley.

For practical purposes, please register prior to 31 January.

Schedule

	Wednesday (12/2)	Thursday (13/2)	Friday (14/2)
10:00 - 11:00	Luzzatto	Galatolo	Lenci
11:30 - 12:30	Marò	Radice	Cardin
14:30 - 15:30	Castorrini	Sorrentino
16:00 - 17:00	Giulietti	Schindler

Location: All talks will be in Aula Dal Passo.
Social: On Thursday evening we'll have dinner together.
Coffee breaks: Each day at 11:00 and at 15:30 served in the area aggregativa.
Lunch: Served each day (including Friday) at 13:00 in the area aggregativa.

Speaker	Title
Franco Cardin (Padova)	Riduzione finito-dimensionale della Termodinamica di Giovanni Jona-Lasinio. Aspetti topologici
Abstract: Muovendo da un ‘toy model’ della termodinamica di non-equilibrio, si propone la costruzione di una riduzione finita esatta dell’equazione degli equilibri, accanto ad una riduzione finito dimensionale approssimata, con stime, dell’equazione dinamica. Si mette in evidenza, tra altri aspetti, l’invarianza dell’indice di Morse degli equilibri dalla versione infinito-dim. a quella finita.
Roberto Castorrini (Tor Vergata)	Statistical properties of partially expanding maps
Abstract: In the last years, an extremely powerful method has been developed to study the statistical properties of a dynamical system: the functional approach. It consists of the study of the spectral properties of the transfer operator on suitable Banach spaces. In this talk I will discuss how to further such a point of view to a class of two dimensional partially hyperbolic systems, not necessarily skew products. To illustrate the scopes of the theory, I will discuss how to apply the results to the case of fast-slow partially hyperbolic systems.
Stefano Galatolo (Pisa)	Linear response in random dynamical systems and control of the statistical properties of dynamics
Abstract: The talk will be quite introductory. We will review the concept of linear response with emphasis to the case of dynamical systems with additive noise. This is an example of a quite general class of random systems for which one has linear response results without assuming hyperbolicity conditions on the deterministic part of the dynamics, thank to the regularizing effect of the noise. We will then discuss the control problem naturally associated to linear response: what is the best perturbation to be applied to a system in order to get some wanted change in its statistical properties? This problem has an evident potential importance in the applications, and still was not much studied.
Paolo Giulietti (SNS)	Quantitative Global-Local mixing for accessible skew-products
Abstract: We study global-local mixing for skew products which are locally accessible. In such framework we prove polynomyal decay of correlations, which is generically optimal, for a reasonably large class of observables. Our strategy is flexible, since it relies on the study of a twisted transfer operator, and could be generalized to many other situations. Joint work with A. Hammerlind and D. Ravotti.
Marco Lenci (Bologna)	Statistical properties of global observables in infinite-measure systems
Abstract: In rough terms, a global observable for an infinite-measure dynamical system is an essentially bounded, non-integrable, function which admits an “infinite-volume average” in some sense. Functions like these represent extensive quantities in extended systems (such as non-compact Hamiltonian systems, etc.). One is interested in the stochastic properties of global observables in infinite-measure-preserving (or similar) dynamical systems: law of large numbers, mixing, limit theorems, etc. In terms of mixing, there exist two categories of definitions: global-global mixing, pertaining to the decorrelation of two global observables, and global-local mixing, which describes instead the decorrelation between a global and a local (i.e., integrable) observable. We report recent results on the mixing properties for two different kinds of infinite-measure systems: interval maps with an indifferent fixed point (in collaboration with C. Bonanno and P. Giulietti) and certain Z^d-extension of chaotic systems and variations thereof (this is work by D. Dolgopyat and P. Nandori). For the former class, it turns out the global-local mixing implies a very unconventional limit theorem that is peculiar to infinite-measure-preserving with one indifferent fixed point. For the latter class, if time permits, we shall discuss how certain forms of global-local mixing may help derive conventional limit theorems (such as the CLT) for global observables.
Mattia Radice (Como)	Some classes of non-homogeneous random walks and their statistical properties
Abstract: I will present two different models of one-dimensional lattice random walk with nearest-neighbour jumps and spatially inhomogeneous transition probabilities, which may be used to describe heterogeneous diffusion, recently considered in many areas of science. In the first model, introduced by Gillis [Gillis, 1956], the probabilities of jumping to the left or right depend on the position of the walker; in the second model, presented in [Artuso, Cristadoro, Onofri, Radice, 2018; Radice, Onofri, Artuso, Cristadoro, 2019] and related to the well-known Lévy-Lorentz gas, a particle initially randomly chooses a direction of motion, and then at each site it can be reflected or transmitted according to a certain probability, which depends on the position along the lattice. I will show how to derive the transport properties of the systems by using appropriate continuum limits, and discuss how to deduce some non-local properties of the processes, such as the survival probability and the occupation time of the positive side of the lattice, by starting from the knowledge of a local property, viz., the long-time behaviour of the probability of occupying the origin.
Alfonso Sorrentino (Tor Vergata)	Inverse problems and rigidity questions in Billiard Dynamics
Abstract: A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain (e.g. the shape of the billiard table): while it is evident how the shape determines the dynamics, a more subtle and difficult question is to which extent the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing unanswered questions and difficult conjectures that have been the focus of very active research over the last decades. In this talk I shall describe several of these questions, with particular emphasis on recent results obtained in collaborations with Guan Huang and Vadim Kaloshin, related to the classification of integrable billiards (also known as Birkhoff conjecture), and to the possibility of inferring dynamical information on the billiard map from its Length Spectrum (i.e., the lengths of its periodic orbits).
Stefano Luzzatto (ICTP)	Hyperbolic coordinates
Abstract: I will define the notion of hyperbolic coordinates and explain how they can be a very useful tool in non-uniformly hyperbolic dynamics.
Stefano Marò (Pisa)	Symplectic methods for twist maps
Abstract: I will describe some techniques to study invariant sets of symplectic twist maps in the framework of Aubry-Mather theory. As an example we will study the dynamics of a bouncing ball problem.
Tanja Schindler (SNS)	trong laws under trimming - comparison between iid random variables and dynamical systems
Abstract: Trimming, i.e. removing the largest entries of a sum of iid random variables, has a long tradition in proving limit theorems which are not valid if one considers the untrimmed sum - for instance a strong law of large numbers for random variables with an infinite mean. For certain ergodic transformations (e.g. piecewise expanding interval maps) and certain observables over these transformations the results are essentially the same as in the iid case. However, considering the same ergodic transformation and an observable with a different distribution function, the system can behave completely different to its iid counterpart. I will give an overview of some of the (sometimes surprising) trimming results in the dynamical systems setting. This is partly joint work with Marc Kesseböhmer.

Chairs: Wednesday morning: Liverani, Wednesday afternoon: Panti, Thursday morning: Bonanno, Thursday afternoon: Carminati, Friday morning: Butterley.

(full schedule including abstracts)

Participants

Alfonso Sorrentino, Alessandro Della Corte, Alessandro Fortunati, Carlangelo Liverani, Carlo Carminati, Claudio Bonanno, Cristian Mendico, Daniele Galli, Franco Cardin, Giampaolo Cristadoro, Gianluigi Del Magno, Giorgio Mantica, Giovanni Panti, Hajar Ravanbod, Jessica Elisa massetti, Manuele Onofri, Marco Lenci, Mattia Radice, Mohammad Khalili Garakani, Mubarak Muhammad, Niloofar Kiamari, Olga Bernardi, Oliver Butterley, Paolo Giulietti, Roberto Artuso, Roberto Castorrini, Sakshi Jain, Sara Munday, Stefano Galatolo, Stefano Isola, Stefano Luzzatto, Stefano Marmi, Stefano Marò , Tanja Schindler,