at the Biological Centre of Roscoff

of the Centre Henri Lebesgue

This conference will address the study of dynamical and stochastic
properties of dynamical systems that exhibit a chaotic (slow or fast)
behaviour, with limit theorems of very different nature.
The aim of this conference is to shed light on the diversity of these
models, their different behaviours, the different methods used
(perturbation of operators, induction, martingales, ...) and unsolved
problems on these questions.

Tuesday 9 | Wednesday 10 | Thursday 11 | Friday 12 | |
---|---|---|---|---|

9h-10h | Sakshi Jain | Richard Aoun | Nguyen-Thi Dang | |

10h-10h30 | Break | Break | Break | Break |

10h30-11h30 | Maria Saprykina | Sara Brofferio | Çağri Sert | Polina Vytnova |

11h30-12h30 | Julien Trévisan | Jean-François Quint | Jérôme Dedecker | Carlos Matheus Silva |

12h45 | Lunch | Lunch | Lunch | Lunch |

14h30-15h | Emmanuel Lesigne | |||

15h-16h | Julien Brémont | |||

16h | Break | Break | Break | |

16h30-17h30 | Stefano Galatolo | Guy Cohen | Hong-Kun Zhang | |

17h30-18h30 | Ian Melbourne | Krzysztof Frączek | Kasun Fernando | |

19h | Diner | Diner | Diner |

Bassam FAYAD, University of Maryland, USA

François Ledrappier , CNRS et Sorbonne Université

Florence MERLEVÈDE, Université Gustave Eiffel, Marne-la-Vallée

Sandro VAIENTI, Université de Toulon

François Ledrappier , CNRS et Sorbonne Université

Florence MERLEVÈDE, Université Gustave Eiffel, Marne-la-Vallée

Sandro VAIENTI, Université de Toulon

Christophe CUNY, Université de Brest

Sébastien GOUËZEL , CNRS et Université de Rennes

Françoise PÈNE, Université de Brest

Barbara SCHAPIRA, Université de Rennes

Sébastien GOUËZEL , CNRS et Université de Rennes

Françoise PÈNE, Université de Brest

Barbara SCHAPIRA, Université de Rennes

Jon AARONSON, Tel Aviv University, Israel

Vincent ALOUIN, Université de Rennes and UBO

Richard AOUN, Université Gustave Eiffel, Marne-la-Vallée

Viviane BALADI, CNRS Sorbonne Université and ITS-ETHZ

Dylan BANSARD-TRESSE, Ecole Polytechnique

Julien BRÉMONT, Université Paris-Est Créteil

Sara BROFFERIO, Université Paris-Est Créteil

Anne BROISE, Université de Paris-Saclay

Jérôme CARRAND, Sorbonne Université

Jean-René CHAZOTTES, Institut Polytechnique de Paris

Guillaume CHEVALIER, Université de Bordeaux

Guy COHEN, Ben-Gurion University, Israel

Jean-Pierre CONZE, Université de Rennes

Yves COUDÈNE, Sorbonne Université

Christophe CUNY, Université de Brest

Nguyen-Thi DANG, Université de Paris-Saclay

Jérôme DEDECKER, Université Paris Cité

Jean-Marc DERRIEN, Université de Brest

Yves DERRIENNIC, Université de Brest

Kasun FERNANDO, Scuola Normale Superiore di Pisa, Italy

Nicholas FLEMING VÁZQUEZ, University of Warwick, England

Krzysztof FRĄCZEK, Nicolaus Copernicus University, Torun, Poland

Stefano GALATOLO, Università di Pisa, Italy

Léo GAYRAL, Université de Toulouse III-Paul Sabatier

Sébastien GOUËZEL, CNRS and Université de Rennes

Jérôme GUÉRIZEC, Nantes Université

Loïc HERVÉ, INSA de Rennes,

Sakshi JAIN, Università degli Studi di Roma "Tor Vergata", Italy

Victor KLEPSTYN, Université de Rennes

Stéphane LE BORGNE, Université de Rennes

François LEDRAPPIER, CNRS and Sorbonne Université

Félix LEQUEN, CY Cergy Paris Université

Emmanuel LESIGNE, Université de Tours

Jialun LI, University of Zürich Zurich, Switzerland

Michael LIN, Ben Gurion University

Hugo MARSAN, Université de Toulouse III-Paul Sabatier

Carlos MATHEUS SILVA SANTOS, CNRS and École Polytechnique

Pierre MATHIEU, Aix-Marseille Université

François MAUCOURANT, Université de Rennes

Ian MELBOURNE, University of Warwick, England

Florence MERLEVÈDE, Université Gustave Eiffel, Marne-la-Vallée

Thi Trang NGUYEN, Université de Bretagne Sud

Marc PEIGNÉ, Université de Tours

Axel PÉNEAU, Université de Rennes

Françoise PÈNE, Université de Brest

Maxence PHALEMPIN, Università degli Studi di Firenze, Italy

Jean-François QUINT, Université de Bordeaux

Emmanuel RIO, Université de Versailles Saint-Quentin, Paris-Saclay

Mikaël ROGER, Lycée Auguste Brizeux, Quimper (CPGE)

Maria SAPRYKINA, Kungliga Tekniska Högskolan, Stockholm, Sweden

Benoît SAUSSOL, Aix Marseille Université

Çağri SERT, Universität Zürich, Switzerland

Virgile TAPIERO, Université de Rennes

Damien THOMINE, Université Paris Saclay

Julien TRÉVISAN, IMJ-PRG

Sandro VAIENTI, Université de Toulon

Polina VYTNOVA, University of Surrey, England

Dalibor VOLNÝ, Université de Rouen

Hao WU, University of Zurich, Suisse

Nasab YASSINE, Université Bretagne Sud

Hong-Kun ZHANG, University of Massachusetts, Amherst, USA

Roland ZWEIMÜLLER, Wien Universität, Austria

Richard AOUN, * Stationary probability measures on the projective space *
Summary

Let (X_i) be a sequence of iid GL_d(R)-valued random matrices (where the dimension d is being fixed once for all). The (non commutative) random walk S_n=X_n ... X_1 on the group GL_d(R) induces a Markov chain on the projective space P(R^d). The goal of the talk is to give a full (qualitative) classification of the stationary probability measures of this Markov chain without any irreducibility assumption, generalizing results of Bougerol--Picard (92) concerning affine random walks and linking the works of Furstenberg--Kifer (82) and those Guivarch--Raugi (07) and Benoist--Quint (16) in the reductive case. Joint work with Çağri Sert.

Julien BRÉMONT,* Random walk in a stratified quasi-periodic environment in the plane *
Summary

We discuss the recurrence properties of the model indicated in the title. The analysis involves former results of Jean-Pierre Conze and Yves Guivarc'h (2000) on random walks in random medium and quasi-invariant measures, that we shall recall precisely.

Sara BROFFERIO,* Random walks on semigroup of matrices: the case of rank one matrices *
Summary

Random products of matrices have been widely studied in the last years for both their theoretical interest and various application. We now have a rich and complete theory for random walks on linear group, however less is known when the matrices are not supposed to be invertible.

The goal of this talk is to present through the exemple of the semigroup of rank one matrices the interest of exploring random walks on semigroups, presenting reasonable generalisation of the group case, new challenges and some results.

Guy COHEN,* On the empirical process sampled along a stationary process *
Summary

Nguyen-Thi DANG* Equidistribution of periodic tori*
Summary

Bowen and Margulis independently proved in the 70s that closed geodesics on compact hyperbolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on PSL(2,R) modulo some discrete cocompact sugroup. In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like SL(d,R) for d>2) and by studying the dynamical properties of geodesic flows in higher rank : the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).

Jérôme DEDECKER* Mixing properties of a class of dynamical systems
*
Summary

We compute the tau-mixing coefficients of a class of dynamical systems. We give some applications to deviations inequalities in case of polynomial or sub-exponential decay of correlations. Based on a joint work with C. Cuny and F. Merlevède

Kasun FERNANDO,* Limit Theorems for a class of unbounded observables with an
application to "Sampling the Lindelöf hypothesis"*
Summary

We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise smooth expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on the real line. The class of observables in the CLT and the MLCLT on the real line include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber (2009) and Steuding (2012) who have proven the Strong Law of Large Numbers for sampling the Lindelöf hypothesis.

Krzysztof FRĄCZEK,* Deviation Spectrum of Ergodic Integrals for Locally Hamiltonian Flows on Surfaces *
Summary

The talk will consists of a long historical introduction to the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surafces as well as some current results obtained in collaboration with Corinna Ulcigrai and Minsung Kim. New developments include a better understanding of the asymptotic of so-called error term (in non-degenerate regime) and the appearance of new exponents in the deviation spectrum (in degenerate regime).

Stefano GALATOLO,* Self Consistent Transfer Operators in a Weak and Not So
Weak Coupling Regime. Invariant Measures, Convergence to Equilibrium, Linear Response *
Summary

We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling. We consider a large class of self consistent transfer operators and prove general statements about existence of invariant measures, speed of convergence to equilibrium, statistical stability and linear response. While most of the results presented are valid in a weak coupling regime, the existence results for the invariant measures we show also hold outside the weak coupling regime. We also consider the problem of finding the optimal coupling between maps in order to change the statistical properties of the system in a prescribed way.

Sakshi JAIN* On dynamics of non-Markov systems *
Summary

We study transfer operators associated to piecewise monotone interval transformations and show that the essential spectrum is large whenever the Banach space bounds L^\infty and the transformation fails to be Markov. Constructing a family of Banach spaces we show that the lower bound on the essential spectral radius is optimal. Indeed, these Banach spaces realise an essential spectral radius as close as desired to be the theoretical best possible case.

Emmanuel LESIGNE*
*

Carlos MATHEUS SILVA,* Non-conical strictly convex divisible sets are highly anisotropic *
Summary

Strictly convex divisible sets X are the universal covers of real projective compact manifolds M. From the dynamical point of view, these objects yield natural examples of topologically mixing Anosov flows. The boundary of a strictly convex divisible set is not very smooth in general: unless X is conical, i.e., an ellipsoid (and, thus, M is a hyperbolic manifold), its boundary is not C^2. In this talk, based on joint work with P. Foulon and P. Hubert, we shall discuss the infinitesimal bends at a typical point of the boundary of a non-conical strictly convex divisible set: in particular, we shall see that these infinitesimal bends are highly anisotropic (i.e., they are distinct in distinct directions) thanks to the features of certain locally constant cocycles over uniformly hyperbolic systems.

Ian MELBOURNE,* Interpretation of stochastic integrals, and the Levy area *
Summary

An important question in stochastic analysis is the appropriate interpretation of stochastic integrals. The classical Wong-Zakai theorem gives sufficient conditions under which smooth integrals converge to Stratonovich stochastic integrals. The conditions are automatic in one-dimension, but in higher dimensions it is necessary to take account of corrections stemming from the Levy area.

In previous work with Kelly, we justified the Levy area correction for large classes of deterministic systems, bypassing any stochastic modelling assumptions.

This talk addresses a much less studied question: is the Levy area zero or nonzero for systems of physical interest, eg Hamiltonian time-reversible systems? In recent work with Gottwald, we classify (and clarify) the situations where such structure forces the Levy area to vanish. Outside of these rare situations, we show that the Levy area is typically nonzero.

Jean-François QUINT,* Local limit theorems for conditioned Birkhoff sums *
Summary

For a centered random walk S_n on R, given x in R, one can estimate the probability that x+S_n belongs to an interval [a,b], while all the x+S_k, 0< k< n, have remained non-negative: it decays as n^{-3/2}. Following a general principle which one can date back to Sinai, this should have an analogue for Birkhoff sums of Hölder continuous observables over hyperbolic dynamical systems. Indeed, this is the case, as follows from our joint work with Ion Grama and Hui Xiao.

Maria SAPRYKINA,* Erratic behaviour for one-dimensional random walks in a generic quasi-periodic environment *
Summary

Çağri SERT,* Counting limit theorems for representations of Gromov-hyperbolic
groups *
Summary

Julien TRÉVISAN,* Limit laws in the lattice point problem *
Summary

In this talk, we are going to take interest in the following general problem. Let L be a unimodular lattice of R^{d}. Let S be a measurable set of finite volume. What is the number N(S,L) of points that belong to both L and S? When S is sufficiently regular, it can be shown that this number N(S,L) is approximated by vol(S) to within an error denoted by R(S,L). The aim of this talk is to present several results that gives us a better idea of the behaviour of R(S,L) in different situations: for example, when S is the united disk dilated by a factor t with t going to infinity and L being a random unimodular lattice or when S is a parallelogram dilated by a factor t with t going to infinity and L being a random unimodular lattice.

Polina VYTNOVA,* A new approach to compute Lyapunov exponents *

Hong-Kun ZHANG,* Hyperbolicity and entropy of convex billiards *
Summary

Billiards are well-known models first introduced by Birkhoff as paradigmatic examples of Hamiltonian systems, and pioneered by Yakov Sinai, Leonid Bunimovich, Nikolai Chernov, etc., as a mathematical model for the Lorenz systems and hard-ball gases. Since then billiards have acquired increasing importance as they shed light in understanding thermodynamic limits, connected to deep issues in quantum and wave physics all the way to quantum chaos. However for some Bunimovich billiards, whose boundary consists of only arcs and straight lines, even the hyperbolicity is not known, not to mention ergodicity and other chaotic properties. In this talk, I will discuss these new classes of convex billiards and their properties. The hyperbolicity for some of them is only numerically proved, which leaves many open questions to explore. I will also review my recent collaboration work with Michal Misiurewicz about topological entropy for Bunimovich stadium.

Let (X_i) be a sequence of iid GL_d(R)-valued random matrices (where the dimension d is being fixed once for all). The (non commutative) random walk S_n=X_n ... X_1 on the group GL_d(R) induces a Markov chain on the projective space P(R^d). The goal of the talk is to give a full (qualitative) classification of the stationary probability measures of this Markov chain without any irreducibility assumption, generalizing results of Bougerol--Picard (92) concerning affine random walks and linking the works of Furstenberg--Kifer (82) and those Guivarch--Raugi (07) and Benoist--Quint (16) in the reductive case. Joint work with Çağri Sert.

Julien BRÉMONT,

We discuss the recurrence properties of the model indicated in the title. The analysis involves former results of Jean-Pierre Conze and Yves Guivarc'h (2000) on random walks in random medium and quasi-invariant measures, that we shall recall precisely.

Sara BROFFERIO,

Random products of matrices have been widely studied in the last years for both their theoretical interest and various application. We now have a rich and complete theory for random walks on linear group, however less is known when the matrices are not supposed to be invertible.

The goal of this talk is to present through the exemple of the semigroup of rank one matrices the interest of exploring random walks on semigroups, presenting reasonable generalisation of the group case, new challenges and some results.

Guy COHEN,

Nguyen-Thi DANG

Bowen and Margulis independently proved in the 70s that closed geodesics on compact hyperbolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on PSL(2,R) modulo some discrete cocompact sugroup. In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like SL(d,R) for d>2) and by studying the dynamical properties of geodesic flows in higher rank : the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).

Jérôme DEDECKER

We compute the tau-mixing coefficients of a class of dynamical systems. We give some applications to deviations inequalities in case of polynomial or sub-exponential decay of correlations. Based on a joint work with C. Cuny and F. Merlevède

Kasun FERNANDO,

We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise smooth expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on the real line. The class of observables in the CLT and the MLCLT on the real line include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber (2009) and Steuding (2012) who have proven the Strong Law of Large Numbers for sampling the Lindelöf hypothesis.

Krzysztof FRĄCZEK,

The talk will consists of a long historical introduction to the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surafces as well as some current results obtained in collaboration with Corinna Ulcigrai and Minsung Kim. New developments include a better understanding of the asymptotic of so-called error term (in non-degenerate regime) and the appearance of new exponents in the deviation spectrum (in degenerate regime).

Stefano GALATOLO,

We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling. We consider a large class of self consistent transfer operators and prove general statements about existence of invariant measures, speed of convergence to equilibrium, statistical stability and linear response. While most of the results presented are valid in a weak coupling regime, the existence results for the invariant measures we show also hold outside the weak coupling regime. We also consider the problem of finding the optimal coupling between maps in order to change the statistical properties of the system in a prescribed way.

Sakshi JAIN

We study transfer operators associated to piecewise monotone interval transformations and show that the essential spectrum is large whenever the Banach space bounds L^\infty and the transformation fails to be Markov. Constructing a family of Banach spaces we show that the lower bound on the essential spectral radius is optimal. Indeed, these Banach spaces realise an essential spectral radius as close as desired to be the theoretical best possible case.

Emmanuel LESIGNE

Carlos MATHEUS SILVA,

Strictly convex divisible sets X are the universal covers of real projective compact manifolds M. From the dynamical point of view, these objects yield natural examples of topologically mixing Anosov flows. The boundary of a strictly convex divisible set is not very smooth in general: unless X is conical, i.e., an ellipsoid (and, thus, M is a hyperbolic manifold), its boundary is not C^2. In this talk, based on joint work with P. Foulon and P. Hubert, we shall discuss the infinitesimal bends at a typical point of the boundary of a non-conical strictly convex divisible set: in particular, we shall see that these infinitesimal bends are highly anisotropic (i.e., they are distinct in distinct directions) thanks to the features of certain locally constant cocycles over uniformly hyperbolic systems.

Ian MELBOURNE,

An important question in stochastic analysis is the appropriate interpretation of stochastic integrals. The classical Wong-Zakai theorem gives sufficient conditions under which smooth integrals converge to Stratonovich stochastic integrals. The conditions are automatic in one-dimension, but in higher dimensions it is necessary to take account of corrections stemming from the Levy area.

In previous work with Kelly, we justified the Levy area correction for large classes of deterministic systems, bypassing any stochastic modelling assumptions.

This talk addresses a much less studied question: is the Levy area zero or nonzero for systems of physical interest, eg Hamiltonian time-reversible systems? In recent work with Gottwald, we classify (and clarify) the situations where such structure forces the Levy area to vanish. Outside of these rare situations, we show that the Levy area is typically nonzero.

Jean-François QUINT,

For a centered random walk S_n on R, given x in R, one can estimate the probability that x+S_n belongs to an interval [a,b], while all the x+S_k, 0< k< n, have remained non-negative: it decays as n^{-3/2}. Following a general principle which one can date back to Sinai, this should have an analogue for Birkhoff sums of Hölder continuous observables over hyperbolic dynamical systems. Indeed, this is the case, as follows from our joint work with Ion Grama and Hui Xiao.

Maria SAPRYKINA,

Çağri SERT,

Julien TRÉVISAN,

In this talk, we are going to take interest in the following general problem. Let L be a unimodular lattice of R^{d}. Let S be a measurable set of finite volume. What is the number N(S,L) of points that belong to both L and S? When S is sufficiently regular, it can be shown that this number N(S,L) is approximated by vol(S) to within an error denoted by R(S,L). The aim of this talk is to present several results that gives us a better idea of the behaviour of R(S,L) in different situations: for example, when S is the united disk dilated by a factor t with t going to infinity and L being a random unimodular lattice or when S is a parallelogram dilated by a factor t with t going to infinity and L being a random unimodular lattice.

Polina VYTNOVA,

Hong-Kun ZHANG,

Billiards are well-known models first introduced by Birkhoff as paradigmatic examples of Hamiltonian systems, and pioneered by Yakov Sinai, Leonid Bunimovich, Nikolai Chernov, etc., as a mathematical model for the Lorenz systems and hard-ball gases. Since then billiards have acquired increasing importance as they shed light in understanding thermodynamic limits, connected to deep issues in quantum and wave physics all the way to quantum chaos. However for some Bunimovich billiards, whose boundary consists of only arcs and straight lines, even the hyperbolicity is not known, not to mention ergodicity and other chaotic properties. In this talk, I will discuss these new classes of convex billiards and their properties. The hyperbolicity for some of them is only numerically proved, which leaves many open questions to explore. I will also review my recent collaboration work with Michal Misiurewicz about topological entropy for Bunimovich stadium.