Program

Dynamics of  Foliations

Talk (September 13, 15h45)
Speaker: Ahmed Elshafei
Title: On completeness of pseudo-Riemannian and holomorphic geodesic flows
Abstract:   This talk is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. Starting by establishing the Euler–Arnold formalism in the holomorphic setting, then focusing on the real Lie group SL(2, R) reobtaining the known characterization of geodesic completeness. In addition, a detailed study of the maximum domain of definition of every single geodesic for every possible metric is presented. Finally, the completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics is investigated and, in particular, establishing a full classification for the Lie group SL(2, C). 

Talk (September 15, 14h00)
Speaker: Miguel Mendes
Title: Representation theorems and ergodic theory
Abstract:  Representation theorems have played a key role in ergodic theory, allowing to relate invariant measures with the space of continuous observables for the dynamics. Moreover, these have been the cornerstone in several approaches to thermodynamic formalism.  Starting with an overview of Riesz Representation Theorem, I will review more general representation theorems and use them to provide new insight on a non-classical thermodynamical formalism framework.  

Minicourse (September 06 and September 08, 14h00-15h30)
Speakers: Yushi Nakano 

Title: Irregular behavior in dynamics
Abstract: These talks aim to review recent progress in the study of irregular behavior of dynamical systems and group actions, which has a good connection to the mini-course "Irregular behavior in foliations" by Yokoyama. 

Part 1: I will consider how large the irregular set (i.e. the set of points whose time average does not exist) for smooth maps is. The positivity of the Lebesgue measure of the irregular set is closely related to the non-hyperbolicity of maps. On the other hand, residual irregular sets are found more universally, especially among hyperbolic maps. More recently, it is even shown that the existence of two distinct invariant probability measures with dense support is enough for the residuality of the irregular set. 

Part 2:  I consider irregular sets for (semi-)group actions. I will see that several sufficient conditions for the residuality of the irregular set for maps can be generalized to group actions, but the criteria may not be very useful when the action is non-Foelber (or non-amenable). On the other hand, a totally different approach allows us to construct irregular sets with positive Lebesgue measure for some class of non-Foelner group actions.

Talk (September 07, 14h00)
Speaker: Ana Rodrigues
Title: Invariant curves for Planar Piecewise Isometries: Recent Advances  
Abstract:  It is believed that the phase space of typical Hamiltonian systems is divided into regions of regular and chaotic motion. Area preserving maps which can be obtained as Poincar{\'e} sections of Hamiltonian systems, exhibit this property as well, with KAM curves splitting the domain into regions of chaotic and periodic dynamics. However, a general and rigorous treatment of these is not yet known. In this talk we will discuss the fact that PWIs, which are area preserving maps,   can exhibit a similar phenomena. 

Talk (September 14, 15h45)
Speaker: Luana Segantim
Title: On the expansiveness of invariant measures under pseudogroups
Abstract: The expansiveness of a measure plays an important role in the study of dynamical systems, providing properties with respect to classical dynamic objects. Based on the work of Arbieto and Morales [Some properties of positive entropy maps, ETDS, 2013], we will define the concept of expansiveness of a measure from the point of view of pseudogroups and then we will discuss a implication in what we will define as the analogue of stable sets for pseudogroups.

Talk (September 13 and September 14, 14h00-15h30)
Speaker: Helena Reis
Title: Meromorphic PSL(2,C)-actions and projective structures on Riemann surfaces
Abstract:  In a fundamental paper A. Guillot linked classical (quadratic) Halphen systems to actions of PSL(2,C). In particular, he was able to clarify the dynamics and the geometry associated with these Halphen systems. In this talk we will consider meromorphic actions of PSL(2,C) and show that they are connected with certain rational vector fields that can be viewed as an extension of Halphen vector fields. I will also comment on some (potential) applications of this construction. This is joint work with A. Elshafei and J.C. Rebelo.

Minicourse (September 06 and September 08, 15h30-16h30)
Speakers: Tomoo Yokoyama 

Title: Irregular behavior in foliations
Abstract: These talks aim to introduce irregular behavior in foliations, which has an excellent connection to the minicourse  "Irregular behavior in dynamics" by Nakano. In particular, we explain the differences and similarities between irregular behavior in dynamics and foliations. 

Part 1. We review foliation theory to define and discuss the irregular set for foliation. In particular, we will review some results on singular foliation on surfaces and codimension one regular foliation. 

Part 2. We introduce some concepts to relate group actions and foliations. For instance, the suspensions of group actions become foliation. Finally, we present some results for irregular behavior for foliation and state the ideas of the proofs.

Markov Structures in dynamical systems

Minicourse (September 25 & September 26, 14h00-15h00)
Speaker: Wael Bahsoun
Title:  Statistical properties of mean-filed coupled systems
Abstract: I will talk about infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical equilibrium. Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions. This is joint work with C. Liverani and F. M. Sélley.  If time permits, I will also talk about analogous results for intermittent systems. The latter is a joint work with Alexey Korepanov. 

Minicourse (September 21, 14h00-15h30, and September 22,  14h00-15:30 & 16h00-17h30)
Speaker: Silvius Klein
Title:  Statistical properties for certain dynamical systems
Abstract:  The main objective of this mini-course is the study of some recent topics in ergodic theory about limit laws (i.e. the large deviations principle and the central limit theorem) for certain types of dynamical systems. The main tool in this study is the existence of the spectral gap of the Markov transition operator or the Ruelle transfer operator in an appropriate space of observables. 

Talk (September 26, 15h30-16h30)
Speaker: Asad Ullah
Title:  Decay of Correlations via Induced Weak Gibbs Markov maps
Abstract: L. S. Young introduced induced Gibbs Markov maps in articles from 1998 and 1999. She showed that the existence of induced Gibbs Markov maps with integrable return time implies the existence of an exact invariant absolutely continuous probability measure with respect to reference measure,  and the rate of decay of correlations is related to the tail of the return time.  In this talk, I will discuss how to obtain similar results under weaker assumptions, which allows the induced map not necessarily to be full branch. This is a joint work with Helder Vilarinho (UBI & CMA-UBI).

Talk (September 25, 15h30-16h30)
Speaker: Odaudu Etubi
Title:  Hölder Continuity of Density and Entropy for Piecewise Expanding Maps
Abstract: In this work, we show the Hölder continuity of invariant densities and entropy for piecewise expanding maps, thereby extending the results in [1, 2]. Furthermore, we built a family of circle maps with neutral fixed points possessing an absolutely continuous invariant measure. Our technique is to show that Lipschitz continuity of the density and entropy for this family of maps possessing a weak Gibbs Markov inducing scheme holds and it is inherited by the density and entropy for the original map. Subsequently, lifting the measures, we show the Hölder continuity of the solenoid map with intermittency. This is a joint work with José Ferreira Alves and Wael Bahsoun.

References:
[1] Alves, J., Pumariño, A., & Vigil, E. (2017). Statistical stability for multidimensional piecewise expanding maps. Proceedings of the American Mathematical Society, 145(7), 3057-3068.
[2] Alves, J. F., & Pumariño, A. (2021). Entropy formula and continuity of entropy for piecewise expanding maps. In Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 38(1), 91-108.

Talk (September 27, 14h00-15h00)
Speaker: João Matias
Title: From Decay of Correlations to Large Deviations in certain Dynamical Systems
Abstract:  A successful approach in Dynamical Systems consists in using inducing schemes with certain recurrence rates to deduce statistical properties such as the existence of invariant measures, decay of correlations or large deviations [5, 6, 7]. In this approach, two types of schemes have been used: Gibbs-Markov structures and Young structures [1]. These structures are highly non trivial and a natural question consists in knowing to what extent this approach can be applied. In [4], it was showed that for systems without contractive directions, certain rates of decay of correlations imply the existence of Gibbs-Markov induced schemes with recurrence rates with essentially the same order. In that proof, the fact that the Perron-Frobenius Operator has good properties in certain Banach spaces played a major role. It was possible to deduce from these properties that, in that context, the decay of correlations implies the large deviations.
Aiming to extend this approach to partially hyperbolic systems with contractive directions of the type considered in [2] and [3], we will consider dynamical systems with certain decay of correlations rates on a Young Tower and show how the previous techniques can be used to obtain the desired large deviations rates. This seminar is based on a work in progress with Professor José Ferreira Alves.

Key Words - Decay of correlations; Large deviations; Young structures; Recurrence rates; Partial hyperbolicity

References:
[1] Alves, J. F. Nonuniformly hyperbolic attractors. Geometric and probabilistic aspects. Springer Monographs in Mathematics. Springer International Publishing, 2020.
[2] Alves, J. F., Bonatti, C., and Viana, M. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140, 2 (2000), 351–398.
[3] Alves, J. F., Dias, C. L., Luzzatto, S., and Pinheiro, V. SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. (JEMS) 19, 10 (2017), 2911–2946.
[4] Alves, J. F., Freitas, J. M., Luzzatto, S., and Vaienti, S. From rates of mixing to recurrence times via large deviations. Adv. Math. 228, 2 (2011), 1203–1236.
[5] Melbourne, I., and Nicol, M. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360, 12 (2008), 6661–6676.
[6] Young, L.-S. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147, 3 (1998), 585–650.
[7] Young, L.-S. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.

Talk (September 27, 15h30-16h30)
Speaker: Vilton Pinheiro
Title: Statistical stability and the geometric center
Abstract: For a reference measure m with an abundance of hyperbolicity, induced Markov maps can be used to prove the statistical stability of m.  For a reference measure m with a poor control of it hyperbolicity, one can mix topological and probabilistic arguments to study the statistical stability of m with respect to generic dynamics. To this end we will introduce the idea of "geometric center" of m and will use it in the study the existence of absolutely continuous invariant measures and also in the study of the statistical stability of m.

Dynamics and Game Theory

Talk (October 13, 15h30)
Speaker: João Paulo Almeida
Title: Hotelling Game on Networks
Abstract: We present a theoretical framework to study the location-price competition in a Hotelling- type network game, extending the Hotelling model with linear transportation costs from a line (city) to a network (town). We show the existence of a pure Nash equilibrium price if, and only if, some explicit conditions on the production costs and on the network structure hold. Furthermore, we prove that the localizations of the local firms are optimal at the cross-roads of the town. In addition, we extend the results to the case where Firms know their production costs but are uncertain about the production costs of the competitor firms.

Talk(October 13, 14h30)
Speaker: Filipe Martins
Title: Characterizing the basins of attraction and global dynamics in a generalized Baliga–Maskin public good model
Abstract: We study an evolutionary dynamics for the contributions by agents to a common/public good in a generalized version of Baliga and Maskin’s environmental protection model. The dynamical equilibria consist of three scenarios: a single agent contributing to preserve the good with its optimal contribution level, and all the other agents being free‑riders: a group of agents with the same optimal contribution level contributing to preserve the good, and all the other agents being free‑riders; one where no agents contribute. We prove that each trajectory converges to the equilibrium associated to the single agent (or group of agents) with the highest preference for the good that are contributing since the beginning. We note that while the aggregate contribution is below the optimal contribution level of the agent with the smallest preference for the good, then the aggregate contribution is increasing and there is no free‑riding. Hence, if the optimal contribution level of the agent with the smallest preference is enough to not exhaust the good too quickly and the optimal contribution level of the agent with the greatest preference is enough to preserve the good, then, in spite of the appear ance of free‑riding in the contributions, the good might not be exhausted.

Talk(October 13, 15h30)
Speaker: João Paulo Almeida
Title: Hotelling Game on Networks
Abstract: We present a theoretical framework to study the location-price competition in a Hotelling- type network game, extending the Hotelling model with linear transportation costs from a line (city) to a network (town). We show the existence of a pure Nash equilibrium price if, and only if, some explicit conditions on the production costs and on the network structure hold. Furthermore, we prove that the localizations of the local firms are optimal at the cross-roads of the town. In addition, we extend the results to the case where Firms know their production costs but are uncertain about the production costs of the competitor firms.

Talk(October 13, 14h00)
Speaker: Alberto Pinto
Title: The power of voting to fight corruption in democratic states
Abstract: We introduce an evolutionary dynamical model for corruption in a democratic state describing the interactions between citizens, government and officials, where the voting power of the citizens is the main mechanism to control corruption. Three main scenarios for the evolution of corruption emerge depending on the efficiency of the institutions and the social, political, and economic characteristics of the State. Efficient institutions can create a corruption intolerant self-reinforcing mechanism. The lack of political choices, weaknesses of institutions and vote buying can create a self-reinforcing mechanism of corruption. The ambition of the rulers can induce high levels of corruption that can be fought by the voting power of the citizens creating corruption cycles.

Talk(October 13, 16h00)
Speaker: Renato Soeiro
Title: Unraveling Paradoxes in Bertrand Competition: The Power of Network Effects
Abstract: Price undercutting dynamics in Bertrand competition models often yield paradoxical outcomes, challenging empirical knowledge. In this talk, we discuss how, in models with assumptions that eliminate these outcomes,  the study of network effects as a perturbation sometimes carries paradoxes of its own. Our central focus is to show how microfounding demand on network effects among buyers/ consumers provides fundamental building blocks for generating positive profit pure price equilibria. Furthermore, we explore how this is sufficient to produce price dispersion and different patterns of individual and group choices while avoiding difficulties provoked by the undercutting dynamics.

Talk(October 13, 16h30)
Speaker: Bruno Oliveira
Title: Convergence of Prices in Imperfect Random Matching Economies with Cobb-Douglas Utility Functions
Abstract: In this research, we investigate imperfect random matching economies where pairs of participants engage in imperfect trade involving two goods,  bounded by Cobb-Douglas utility functions. Our study centers on the con- vergence of prices in these economies to the Walrasian price under specific  symmetry conditions, which depend on both the initial distribution of endowments and agents’ preferences. In such economies, we show that the  expectation of the logarithm of trading prices at each period is shown to  be equal to the logarithm of Walrasian prices. This convergence offers in- sights into the self-organization of agents in seemingly disorganized mar- kets. Moreover, this results extends to economies with multiple goods, and  under the appropriate symmetry conditions, convergence to Walrasian prices remains a robust result.

Talk(October 13, 17h30)
Speaker: Miguel Arantes
Title: Revolutionizing Demand Forecasting in Fashion Retail: Harnessing the Power of LSTM for Enhanced Accuracy and Interpretability
Abstract: This research is dedicated to elevating demand forecasting practices in the fashion retail industry through an in-depth exploration of Long Short-Term Memory (LSTM) techniques. Our primary goal is to develop innovative and interpretable approaches to tackle the industry's demand forecasting challenges, relying solely on LSTM. By leveraging LSTM's capabilities, we aim to provide more robust and dependable solutions while also contributing to the advancement of the field of demand forecasting.

Topological methods in dynamics

Minicourse ( 2 meetings, TBA )
Speaker: Nilson Bernardes Jr
Title:  Some recent contributions to Linear Dynamics
Abstract: Expansivity, hyperbolicity, structural stability and the shadowing property are fundamental concepts in the qualitative theory of dynamical systems and differential equations, with many applications in Mathematics, Physics and other branches of Science. In the last few years some important achievements were obtained in the setting of linear dynamics, including the solution of a long-standing open problem. Below we list some of these achievements:

In our short course, we will give an overview on this subject. We will recall the relevant concepts, present some proofs and counterexamples in detail, and discuss some basic questions on these notions for linear operators that are still open.  Our exposition will be based on some recent papers of the speaker with various collaborators, namely, Patrícia R. Cirilo, Udayan B. Darji, Ali Messaoudi, Alfredo Peris and Enrique R. Pujals.  

Talk(TBA)
Speaker: Everaldo Bonotto
Title: Recursiveness properties on impulsive dynamical systems
Abstract: In this talk, we present the study of time-varying impulsive recursive motions. We investigate the connections among minimal sets, non-wandering points, the Birkhoff center, and attractors. As an illustration, the obtained results are applied to a semilinear reaction-diffusion equation with impulses.

Talk(TBA)
Speaker: Ariel Caticha
Title: Probability, Entropy, and Information  Geometry
Abstract: A main concern of any theory of inference is to pick a probability distribution from a set of candidates and this immediately raises many questions. What if we had picked a neighbouring distribution? What difference would it make? What makes two distributions similar? To what extent can we distinguish one distribution from another? Are there quantitative measures of distinguishability? Our goal here is to address such questions by introducing methods of geometry. More specifically the goal will be to introduce a notion of “distance” between two probability distributions. To educate our intuition I will briefly sketch a couple of derivations of the information metric and discuss a couple of examples. I will not develop the subject in all its possibilities, but I will emphasize one specific result – the extension of the Method of Maximum Entropy to discuss fluctuations (or large deviations). Time permitting, I will briefly mention the centrality of information geometry to the foundations of quantum mechanics.

Talk(TBA)
Speaker: Rodolfo Collegari
Title: Attractors for impulsive dynamical systems
Abstract: In this talk we will see the notion of impulsive dynamical systems, the concept of global attractor and some results that guarantee the existence of such object.

Talk(TBA)
Speaker: Pedro Matias
Title: Positive topological entropy for Semi-Riemannian geodesic flows
Abstract: We generalize a Contreras theorem to the case of semi-Riemannian metrics. More precisely, we prove that a smooth semi-Riemannian metric with a non-hyperbolic closed geodesic or with an infinite number of closed geodesics, can be C2-perturbed to a metric whose geodesic flow has positive topological entropy.

Talk(2 meetings)
Speaker: Piotr Oprocha
Title: On tracing properties, invariant measures, and entropy.
Abstract: In 1970s Rufus Bowen related hyperbolic dynamics with specification  property and used this approach to show that there exists a unique measure of maximal entropy. Almost the same time Karl Sigmund used specification property as a tool in characterization of simplex of invariant measures. Since then, these results were inspiration for numerous mathematicians in various studies of dynamics. Several weaker versions of specification property were developed and used as a tool for better understanding of dynamics. At the same time, questions, how often such properties can be found in dynamics were raised (e.g. in the sense of Baire category theorem). In this talk we will present selected results fitting into the above framework of research.

Talk(TBA)
Speaker: Gustavo Pessil
Title: On the classical variational principle for the metric mean dimension
Abstract: Metric mean dimension is a geometric invariant of dynamical systems (X, d, T) with infinite topological entropy. It quantifies the rate at which the amount of ε-distinguishable orbits goes to infinity as ε → 0. As in the topological pressure of finite entropy systems, one can add the dependence on a continuous potential φ: X → R. Being a renormalization of the entropy, which now depends on the choice of equivalent metric to generate the topology, it is natural to search for a measure-theoretic notion of metric mean dimension H satisfying the classical variational principle, namely  mdimM(X, d, T, φ) = sup_μ { H(μ) +  \int φ dμ }.  In this talk we will define such an object, state the variational principle and compute it explicitly for some classical examples of the theory. On these examples the obtained formula for the metric mean dimension with potential will be given in terms of ergodic optimization.

Talk(October 20, 16h00)
Speaker: Mark Pollicott
Title: Homology and entropy for Anosov flows
Abstract: Given an invariant measure for a flow f we can associate to it its asymptotic cycle.  This is a useful notion introduced by Schwartzman and is an element in the first homology measuring the average "drift" in homology of a typical orbit for an f-invariant probability measure.    Now, as is well known, if we take the supremum of the entropies over all possible f-invariant probabilities we obtain the topological entropy. However, if we restrict the supremum to measures whose asymptotic cycle takes a given value c, say, then we obtain the so-called homologically constricted entropy h(c).   For Anosov flows,  this value has been seen to be useful in closed orbit counting and equidistribution  problems.  We will describe the dependence on h(c) on the underlying Anosov flow.  We will also describe some applications of this result.

Talk (TBA)
Speaker: Paulo Varandas
Title: Gluing type properties and their impact on irregular sets
Abstract: In this talk I will recall and compare several topological properties, concerning reconstruction of orbits in dynamical systems, as specification, the gluing orbit property and non-uniforrm versions  of the latter, among others. I will discuss the classes of dynamical systems which satisfy these type of properties and  illustrate one of their simple applications, which is to show that the set of points which fail to satisfy an ergodic theorem is topologically large.

Dynamics of singularities and networks

Preliminary schedule


I. Mini-courses and tutorials


02/Nov (Thursday): Sofia Castro (FEP, CMUP)

Subject: Heteroclinic dynamics 


03/Nov (Friday): Manuela Aguiar (FEP, CMUP)

Subject: Coupled cell networks I


06/Nov (Monday): Ana Paula Dias (FCUP, CMUP)

Subject: Coupled cell networks II


07/Nov (Tuesday): Liliana Garrido-da-Silva (FCUP, CMUP)´

Subject: Stability of cycles and networks


09/Nov (Thursday): Santiago Ibañez (University of Oviedo)

Subject: Unfolding of singularities (nilpotent, Hopf-zero, Hopf-Hopf)


10/Nov (Friday): Alexandre Rodrigues (ISEG, CMUP)

Subject: Chaotic dynamics near a heteroclinic network



II. Talks


11/Nov (Monday): Alexander Lohse (Hamburg University)

 

12/Nov (Tuesday): Chris Bick (Amsterdam University)


13/Nov (Wednesday): Soeren von der Gracht (Paderborn University)


14/Nov (Thurday): Haibo Ruan (Hamburg University)


15/Nov (Friday): Telmo Peixe (Lisbon University)



Probabilistic methods in dynamics

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, 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: TBA

Abstract: TBA


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.).