Workshop on Probabilistic methods in dynamics

Scientific committee of the workshop:

Jorge  Freitas

Mike Todd

Sandro Vaienti

Campus map with rooms and buildings location

Minicourse(December 11, 12, 13, 10h00)

Speaker: Sandro Vaienti

Title: Spectral approach to Poisson statistics in dynamical systems

Abstract: We will introduce a spectral technique to study the convergence to compound Poisson distribution for dynamical systems. The mini-course will include

- Proof of the perturbative result by Keller-Liverani (KL)

- A brief remind of infinite divisible laws and compound Poisson

- A simple application of KL to 1-D deterministic maps

- A (less) simple application to 2-D maps with anisotropic Banach spaces

- Extension of KL to non-stationary situations: sequential and quenched perturbed dynamical systems.

The material is taken from the three articles

https://www.mat.uniroma2.it/~liverani/Lavori/live0802.pdf

https://arxiv.org/pdf/2307.00774.pdf

https://arxiv.org/pdf/2308.10798.pdf

Talk(December 11, 11h30)

Speaker: Mark Holland

Title: On Recurrence, hitting and almost sure growth of extremes

Abstract: We take a measure preserving dynamical system, and a sequence of shrinking targets (B_n), which we take to be nested sets. This talk will overview recent results concerning infinitely often visits to such sets. It is natural to take (B_n) to be a sequence of shrinking balls, but more general sets can be considered too. We highlight connections to quantitative recurrence statistics and extreme value theory.

Talk(December 11, 15h00)

Speaker: Jean-René Chazottes

Title: Large deviations for hitting and return times for equilibrium states on the full shift

Abstract: The context is the full shift on a finite number of symbols equipped with an equilibrium state for a potential with summable variation. I will present recent results from two papers which deal with fluctuations of first return times to cylinder sets, as well as the time it takes to see the first n symbol of a sequence in another sequence (the so-called waiting time). These times grow almost everywhere like exp(n h), where h is the entropy of the measure. What happens at the scale of the central limit theorem is rather obvious because it boils down to the central limit theorem for the Birkhoff sum of minus the potential. In contrast, what happens at the level of large deviations is rather unexpected. In particular, non-convex rate functions show up. Notice that the case of product measures is already non-trivial.

Talk(December 11, 16h30)

Speaker: Mike Todd

Title: Continuity of measures in nonuniformly hyperbolic settings

Abstract: Given a `good' dynamical system with an equilibrium state $\mu$, the distance of an invariant measure $\mu'$ from this can be expressed via the so-called EKP inequality, which gives the difference in terms of a square root of a difference of pressure functions. We consider cases where the system is subexponentially mixing and show that in many such cases the square root must be replaced by a power related to the mixing rate. This is joint work with Iommi and Terhesiu.

Talk(December 12, 11h30)

Speaker: Jiakang Wang

Title: Mixture of two specific maps

Abstract: We consider the maps $T_1$ which is $2x$ when $x < \frac{1}{2}$ and $2-2x$ when $x > \frac{1}{2}$, $T_2=\frac{2x}{x+1}$  on the interval $[0,1]$. We mix them evenly with probability $(\frac{1}{2},\frac{1}{2})$. Will there be an absolutely continuous invariant measure? If so, what will it be?

We would use a bounded variation argument to consider the existence of this invariant measure. In details, we construct a structure to separate the cases and do estimate.

Why is this interesting? This is because it is a critical case, the derivative of $T_2$ is exactly 2 on the right end point while the absolute value of the derivative for $T_1$ is $\frac{1}{2}$. Also, if one assigns the probability not as $(\frac{1}{2},\frac{1}{2})$, then the limiting results will fall into one of these two functions.

Talk(December 12, 12h00)

Speaker: Alexander Baumgartner

Title: Complex Continued Fractions, Extreme Value Theory and Cusp Excursions

Abstract: There is a well-known correspondence between the geodesic flow on the modular surface and continued fractions, first noticed by Artin and then later expanded upon by Adler and Flatto, and Series. The topic of this talk is about a similar connection for three-dimensional analogues of the modular surface and complex continued fractions. We prove a result on the limiting distribution of cusp excursions on these analogues. 

Talk(December 12, 15h00)

Speaker: Alexei Korepanov

Title: Skew product fast-slow systems driven by superdiffusive dynamics such as billiards with flat cusps

Abstract: I will talk about limiting behavior of differential equations of the type dX_n = A(X_n) dt + B(X_n) dS_n, where S_n is a scaled Birkhoff sum process converging to an alpha stable Lévy process in the sense of finite dimensional distributions but not in any of the Skorokhod topologies. We observe such processes in simple and very hands-on examples of billiards with flat cusps. In a joint work with Ilya Chevyrev and Ian Melbourne we found a surprisingly natural approach to the convergence of both S_n and X_n.

Talk(December 12, 16h30)

Speaker: Ian Melbourne

Title: Statistical and non-statistical behaviour for intermittent maps  

Abstract: Consider an intermittent map with two fixed points with the same neutrality in the regime where the absolutely continuous invariant measure is infinite.  Empirical measures do not converge weak*; the limit points are the convex combinations of the Dirac deltas at the fixed points. Nevertheless, the pushforwards of Lebesgue measure (or any absolutely continuous probability measure) converge weak* to a specific convex combination of the Dirac deltas.  (This is joint work with Coates, Luzzatto and Talebi.)

Talk(December 13, 11h30)

Speaker: Benoît Saussol

Title: Quantitative recurrence for the $T,T^{-1}$ transformation

Abstract: We look at the behavior of return times within small regions for the $T,T^{-1}$-transformation. We find diverse asymptotic behaviors, characterized by distinct scaling and limit point processes, depending upon the dimensions of the measures associated with two underlying dynamical systems. The observed behavior resembles that of the direct product of these systems in some cases, mirrors a $\mathbb Z$- extension of the driving system, or takes on a more intricate process. This is a joint work with Françoise Pène.

Talk(December 14, 10h00)

Speaker: Stefano Galatolo

Title: Rare Events and Hitting Time Distribution for Discrete Time Samplings of Stochastic Differential

Equations

Abstract: We consider a random discrete time system in which the evolution of a stochastic differential equation is sampled at a sequence of discrete times. We set up a functional analytic framework for which we can prove the existence of a spectral gap and estimate the behavior of the leading eigenvalue of the related transfer operator as the system is perturbed by putting a ”hole” in it that corresponds to a rare event. By doing so, we derive the distribution of the hitting times corresponding to the rare event and the extreme value theory associated with it.

Talk(December 14, 11h30)

Speaker: Giorgio Mantica

Title: A dynamical model for epidemics 

Abstract: Since the original works of Daniel Bernoulli to the present day, systems of ordinary differential equations have been used to simulate the number of infected, recovered, and diseased people during an epidemic. After briefly reviewing the interesting history of this approach and outlying its power and limitations---specifically, any single individual is only described as belonging to one of such categories---I will introduce a different paradigm. Firstly, will I employ a chaotic one-dimensional map to simulate the internal health dynamics of each person. Then, I will introduce a deterministic (non-probabilistic) rule by which different individuals act upon each other when transmitting an infection. I postulate that each dynamical map is a node in a complex network: in so doing, the resulting full dynamical system is a coupled-maps network amenable to theoretical analysis. At a crossroad between probability and dynamical systems theory, I will show that a phase transition occurs, by which an initial contagion can affect a large size of a population. Dynamics provide the quantitative ingredients to a probabilistic analysis modeled after a previous work by Ott et al., which is validated in this fully deterministic system. Finally, time permitting, I will mention the formal analogy between this problem and the statistics of cascades of neural excitations in the cerebral cortex. This material has been presented in enlarged, educational form aimed at graduate students (comprising twenty guided problems) in the American Journal of Physics. 

Talk(December 14, 15h00)

Speaker: Boyuan Zhao

Title: Maximal length of subsequence matching and Renyi entropy

Abstract: Given a pair of points (x,y) in a subshift system, one can investigate their first n symbols and determine the length of the longest common substring M_n(x,y). Under good mixing conditions, one can prove that there is a log limit rule between M_n and the Renyi entropy of the system for typical pairs of (x,y). This log rule can be generalised to any typical collection of k-points for k greater than 2. For k=1, the convergence remains valid almost everywhere with respect to a Gibbs measure. 

Talk(December 14, 15h30)

Speaker: Dylan Bandard-Tresse

Title: Inducing techniques for hitting time statistics and  application to Misiurewicz and doubly intermittent maps 

Abstract: The induction approach has proven fruitful to get hitting and return time statistics results for non uniformly hyperbolic systems. However, it requires the asymptotically rare events to lie inside the basis on which one induces and it is not always possible to induce on an arbitrarily set and have good mixing properties for the return map. The topic of this talk would be to consider generalized inducing results when we allow the target sets to be outside the basis. We will apply this approach to Misiurewicz and doubly intermittent maps. 

Talk(December 14, 16h30)

Speaker: Nicolai Haydn

Title: Return times distribution and the compound Poisson distribution

Abstract: We consider a deterministic dynamical system with an invariant measure and look at the limiting distribution of return and entry times to metric neighbourhoods of zero measure sets. We show that the limiting distribution generally is compound Poisson distributed and provide a method that allows us to find the associated parameters. We then apply this to the standard parabolic map on the interval and in particular to returns near the parabolic point which requires a non-standard scaling that previously had been found in a way that required considerably more explicit computations. This is joint work with S. Vaienti.

Talk(December 15, 11h30)

Speaker: Françoise Pène

Title: Mixing rate for natural observables for the periodic Lorentz gas flow with infinite horizon

Abstract: The periodic Lorentz gas describes the behavior of a point particle moving at unit speed and bouncing according to the Descartes reflection law off Z^d-periodically arranged "round" obstacles. We consider both the Z^2-periodic Lorentz gas in the plane and the Z-periodic Lorentz gas on a tube. We are interested in the question of mixing rate (in infinite measure) for the flow and for regular observables with compact support.

This model behaves very differently depending on whether the horizon is finite or infinite. While when the horizon is finite an expansion of every order has been established (for the flow) in a previous work in collaboration with Dmitry Dolgopyat and Péter Nandori, simply obtaining the dominant term (for the flow) when the horizon is infinite is an extremely difficult question due to the fact that the time until the next collision is not integrable with respect to the Lebesgue measure preserving the flow (the time between 2 collisions is not square integrable with respect to the invariant measure for the transformation). The aim of this talk is to explain the proof of the infinite horizon mixing result. The proof of this result, established in collaboration with Dalia Terhesiu, relies on precise estimates in probabilistic limit theorems and on a novel tension criterion, and is carried out in two steps. The first step consists in study observables with compact support for the suspension flow (i.e. whose support consists of configurations that are at bounded time away from the nearest reflection time). The second step consists in extending the result to observables with compact support in the space of configurations (such a compact can contain configurations whose trajectory will never encounter an obstacle).

For the first step, we show and use a mixing local limit theorem (MLLT) for the collision application and for the pair (cell number, collision time), we further use a subtle large local deviation (LLD) result obtained in a previous work in collaboration with Ian Melbourne and Dalia Terhesiu. For the second step, we prove and use a novel "tightness" argument (whose proof is non-trivial, but uses precise error terms in MLLT, LLD, but also a large deviaton estimate for the number of collisions, etc.).