Relatori

Lorentz introduced the well known Lorentz gas, a fundamental model in Statistical Physics. Sinai studied its statistical properties in the periodic case. The non periodic case remains a challenge. I will illustrate how ideas from Hilbert and Birkhoff (Garrett) may help in studying the latter problem.

In this talk, we will present some new results regarding the spectrum of transfer operators associated with different classes of dynamical systems. Our goal is to obtain precise information on the discrete spectrum. We will describe a general principle which allows us to obtain substantial spectral information. We will then consider several settings where new information can be obtained using this approach, including affine expanding Markov maps, monotone maps, hyperbolic diffeomorphisms. (Joint work with: Oliver Butterley & Carlangelo Liverani.)

Trapping problems have a long-standing interest in the physics community and beyond: they have been studied with various dynamics for the particle and in a wide variety of static, dynamic and random environments. Nevertheless, traps are generally represented by point absorbers, either on a lattice or in continuous space, and their spatial extent is usually neglected. We try to answer the following natural question: "How is the survival probability of a particle affected in the presence of traps with a finite size?". We consider one-dimensional discrete-time random walks (RWs) in the presence of finite size traps of length l (over which the RWs can jump), periodically distributed and separated by a distance L. We obtain exact results for the mean first-passage time and the survival probability in the special case of a double-sided exponential jump distribution. While such RWs typically survive longer than if they could not leap over traps, their survival probability still decreases exponentially with the number of steps. Its decay rate depends in a non-trivial way on the trap length l and exhibits an interesting regime when l tends to 0 as it tends to the ratio l/L, which is reminiscent of strongly chaotic deterministic systems. Then we generalize our model to continuous-time RWs, where we introduce a power-law distributed waiting time before each jump: in this case, the survival probability decays algebraically with an exponent that is independent of the trap length. Finally, we derive the diffusive limit of our model and show that, depending on the chosen scaling, we obtain either diffusion with uniform absorption, or diffusion with periodically distributed point absorbers. It would be interesting to pursue further the connection with the classical billiard problems and investigate to what extent there are universal features with arbitrary jump distributions. (Joint work with: Benjamin De Bruyne)

To understand the local behaviour of a dynamical system near a fixed point, a first step are local changes of coordinates that result in a simpler form of the system. If the system in question is an iterated holomorphic function, we can represent all of the involved mappings as convergent power series. Problems such as full linearisation or flattening of stable manifolds can then be split into two parts: 1. The formal solution of an equation of power series. 2. The convergence of the formal solution. The expression of such a formal solution naturally contains a series of divisors, that form an obstruction to general convergence when they are very small, making it a small divisor problem. These divisors depend only on the linear part of the map at the fixed point, hence one can formulate sufficient conditions for convergence purely in terms of this linear part. This talk will cover a series of simplification results under Brjuno type conditions on eigenvalues at fixed points, starting with the foundational linearisation results by Siegel and Brjuno, and continuing with more general simplification results by Pöschel and myself, that will enable the complete classifi- cation of the stable dynamics near certain so-called “one-resonant” fixed points under a Brjuno type condition.