cherubino

Research

Statistical Properties of dynamical Systems

Our group is interested in the study of the statistical properties of discrete and continuous group actions describing physical phenomena at different scales. We are mostly interested in spectral methods, which have interesting applications in other contexts. The dynamical systems we study are divided into three classes: uniformly hyperbolic systems; non-uniformly hyperbolic and infinite-measure systems; parabolic systems

Uniformly hyperbolic systems. We are interested in the statistical properties of single and coupled systems. The main properties we study are the existence of rare events and their frequency, and the robustness of the system to small perturbations.

Non-uniformly hyperbolic and infinite-measure systems. In this case, many of the classical results fail. We are interested in studying the asymptotic behaviour of Birkhoff sums (i.e. the sum of an observable along the orbits of the system), the notion of mixing (i.e. the speed of the loss of memory along the orbits of the system), the analytical properties of the transfer operator and the dynamical zeta function (tools of the so-called thermodynamic formalism approach). In particular, we consider dynamical systems related to regular and multi-dimensional continued fractions algorithms, and to the geodesic and horocycle flows on modular surfaces.

Parabolic systems. These systems are characterised by slow convergence for the Birkhoff sums of the observables along the orbits of the system. The main idea is to accelerate the system or decompose it by looking at observables in suitable anisotropic spaces of functions. By these techniques, one obtains information on the speed of convergence to the equilibrium state and identifies eventual deviations from the typical behaviour or obstructions to it.

Members:
  • Claudio Bonanno
  • Carlo Carminati
  • Roberto Castorrini
  • Stefano Galatolo
  • Paolo Giulietti
Collaborators:

Stochastic dynamical systems and applications

Stochastic Dynamical Systems for Climate Studies. We are interested in applying transfer operator techniques and results obtained in the case of abstract dynamical systems to certain climate models. Once a model flow has been constructed, it is possible to construct operators close in spirit to the transfer operator associated to the time-one map of the flow. The statistical properties of such a system can then be investigated through the functional analytic properties of the operator. We are particularly interested in three aspects of such a study. First, when we interpret the response of the system to the perturbations, we investigate how the invariant measure changes, i.e., the so-called “linear response” of the system (or its absence). Second, we compare our abstract asymptotic results with the existence of appropriate time scales, we do so by scouting for metastable observables, i.e., for example, the system might not be in equilibrium but might stay very close to a periodic cycle for a very long time. Last, we study the appearance and distribution of extreme climatic events. This can be done by exploring recurrence properties to special sets of the phase space on which chosen observables are above/below certain thresholds, which encode the extreme event.

Thermodynamic Formalism, Stochastic Filter and Stochastic Billiards. We study the ergodic properties of stochastic dynamical systems generated by transformations with hyperbolic behaviour on average through the analysis of the spectrum of the transfer operator (thermodynamic formalism). This approach has interesting applications to the problem of the stability of the stochastic filter in probability theory. The problem consists in obtaining an optimal estimate of the state of the system starting from a sequence of observations affected by noise and showing that such an estimate loses memory of its initial condition. We are also interested in the study of stochastic billiards that are obtained from a deterministic billiard by replacing the law of specular reflection with a stochastic reflection law. An orbit of a stochastic billiard table is a Markov chain with transition probability depending on the geometry of the billiard table and the law of reflection. The goal is to show the existence of stationary measures for a large family of billiards with stochastic reflection. Of particular interest is the comparison between the statistical properties of a stochastic billiard and those of the corresponding deterministic billiard.

Members:
  • Gianluigi Del Magno
  • Stefano Galatolo
  • Paolo Giulietti
  • Carlo Carminati
  • Francesco Grotto
  • Leonardo Roveri
Collaborators:
  • J. Bröcker (Univ. Reading, UK)
  • J. Gaivão, J. Lopes Dias (ISEG – Univ. Lisbon, Portugal)
  • S. Vaienti (Centre de Physique Théorique – Marseille, Francia)

Dynamical systems with singularities and billiards

Hyperbolic Systems with Singularities. A hyperbolic dynamical system is characterized by the presence of expanding and contracting directions. This property produces a complex behavior that in many ways appears stochastic despite the deterministic nature of the system. We are interested in hyperbolic dynamical systems with piecewise regular dynamics (systems with singularities). They represent natural models for mechanical systems with collisions such as billiards. The main aim of the project is the study of the ergodic properties of hyperbolic systems with singularities. In particular, we want to study the existence and properties of special invariant measures called SRB that describe the statistical properties of most of the system’s orbits. We are also interested in the construction of hyperbolic billiards and the analysis of their ergodic properties. Conditions on the geometry of the billiard table and on the law of reflection that guarantee the hyperbolicity of billiard tables in the plane are well known. The goal is to weaken these conditions and extend them to billiards in Euclidean spaces of any dimension.

Generic Properties of Convex Billiards. We investigate generic properties of billiards in convex regions of any size with a regular border (Birkhoff billiards). A property of particular interest is the positivity of the topological entropy which guarantees the existence of a non-trivial hyperbolic set (Smale’s horseshoe). The genericity of this property is known for Birkhoff billiards in the plane. The extension of this result to Birkhoff billiards in arbitrary dimensions is the main objective of this project.

Members:
  • Gianluigi Del Magno
Collaborators:
  • M. Bessa (Univ. Beira Interior, Portugal)
  • P. Duarte (Univ. Lisbon, Portugal)
  • J. Gaivão, J. Lopes Dias (ISEG – Univ. Lisbon, Portugal)
  • M. Torres (Univ. Minho, Portugal)
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