Dynamical Systems, Statistical Properties and their Applications
There will be a conference at CIRMLuminy, Marseille(s), France, in the general area of Dynamical Systems, Statistical properties and their Applicationsfrom 913 December, 2019. 

Participants will include: W. Bahsoun, P. Berger, A. Bufetov, J. Buzzi, J.P. Chazottes, V. Climenhaga, J.P. Conze, D. Dragicevic, K. Falconer, T. Fisher, J. Fraser, S. Gouezel, C. Guillarmou, M. Holland, X. Jin, Y. Kifer, V. Kleptsyn, G. Knieper, F. Ledrappier, S. Marmi, C. Matheus, I. Melbourne F. Naud, M. Nicol, A. Nogueira, M. Peigne, F. Pene, H. Rugh, B. Saussol, S. Seuret, R. Sharp, K. Simon, M. Todd, P. Varandas, E. Verbitskiy, H. Zhang, M. Zworski. 
If you are interested in attending, or for more information, please contact Mark Pollicott (masdbl@warwick.ac.uk)
The meeting will start on the morning of Monday, 9 December and end by lunch time on Friday, 13 December.
Meals and most accommodation will be onsite. There is some limited support available from CIRM, EPSRC and NSF.
Timetable
Monday  Tuesday  Wednesday  Thursday  Friday  
8:509:00  Opening 

9:009:40  Berger  Knieper  Saussol  Jin  
9:5010:30  Takahasi  Climenhaga  Todd  Fraser  Nogueira 
10:5511:35  Marmi  Fisher  Holland  Falconer  Melbourne 
11:4012:20  Short Talks I  Peigne  Varandas  Short talks IV 
Buzzi 
Lunch interval  
14:0015:55  Short talks II 
Short talks III 
Short talks V 

16:0016:40  Kifer  Bufetov  Free afternoon  Seuret  
16:5017:30  Kleptsyn  Naud  Verbitsky  
17:5518:35  Conze  Zworski  Gouezel  
18:4519:25  Ledrappier  
Dinner 
Bouillabaisse  UK Election 
Short talks
Short talks I 
Short talks II  Short talks III  Short talks IV  Short talks V 
Nakano(11:4012:05) 
Cantrell(14:0014:25) Colognese(14:3014:55) Selley(15:0015:25) Su(15:3015:55) 
GarciaZelada(14:0014:25) Crimmins(14:3014:55) Doan(15:0015:25) Berezin(15:3015:55) 
Stadlbauer(11:4012:05)  Fernadno(14:0014:25) Cioletti(14:3014:55) Castorrini(15:0015:25) Yang(15:3015:55) 
Titles and Abstracts (40 minute talks)
A. Bufetov, Determinantal point processes and systems of reproducing kernels.
V. Climenhaga, Closed geodesics and the measure of maximal entropy on surfaces without conjugate points
For negatively curved Riemannian manifolds, Margulis gave an asymptotic formula for the number of closed geodesics with length below a given threshold. I will describe joint work with Gerhard Knieper and Khadim War in which we obtain the corresponding result for surfaces without conjugate points by first proving uniqueness of the measure of maximal entropy and then following the approach of recent work by Russell Ricks, who established the asymptotic estimates in the setting of CAT(0) geodesic flows.
JP. Conze, Step cocycles over rotations: temporal and spatial limit theorems
In the recent years, there has been a growing interest in a distributional limit theorem for ergodic sums over dynamical systems with zero entropy. For rotations x > x+ a on the circle, the pioneer result of J. Beck (2010) on a ``temporal'' limit theorem (for the ergodic sums of special step functions over some quadratic rotations) has opened a new field leading to the results of D. Dolgpayt and O. Sarig, M. Bromberg and C. Ulcigrai.
For the ``spatial'' point of view, under analogous conditions, an asymptotic normal behaviour holds when the time is restricted to some sequence of density 1 (JP. C. and S. Le Borgne).
The aim of the talk is to give a brief overview of these different results, with emphasis on Diophantine conditions for aand for the discontinuity points of the step function.
K. Falconer, Intermediate dimensions, capacities and projections
The talk will review recent work on intermediate dimensions which interpolate between Hausdorff and box dimensions. We relate these dimensions to capacities which leading to ‘Marstrandtype’ theorems on the intermediate dimensions of projections of a set in R^n onto almost all mdimensional subspaces. This is collaborative work with various combinations of Stuart Burrell, Jonathan Fraser, Tom Kempton and Pablo Shmerkin.
S. Gouezel, An approximate Livsic theorem
The Livsic theorem in hyperbolic dynamics ensures that a Hölder function
whose average vanishes along periodic orbits has to be a coboundary. I
will discuss the situation where one only knows that the average is
small, along periodic orbits of large but finite length: can one deduce
that the function is close to being a coboundary, in a quantitative
sense? Motivations and applications will also be discussed. Joint work
with T. Lefeuvre.
I will discuss the problem of determining the existence (or otherwise) of an almost sure growth rate for the maximum of a time series of observations, as generated by a dynamical system. In the context of extreme value theory, such results are analogous to having a `strong law of large numbers', where now we look at maxima, rather than sums. Various dynamical system examples will be considered, such as hyperbolic systems, or nonuniformly expanding interval maps. The main techniques used to determine the behaviour of maxima involve Strong Borel Cantelli results for certain shrinking target sequences.
X. Jin, Mandelbrot cascades acting on ergodic measures.
Abstract: We consider Mandelbrot cascades acting on ergodic measures on a finite alphabet symbolic space. We show that the associated Mandelbrot martingale is degenerate/nondegenerate if the entropy of the random cascades is strictly smaller/greater than the entropy of the ergodic measure. In the critical case when the two entropies are equal, under an extra Gibbslike condition we show that the associated martingale is also degenerate. The proof uses the change of measures method in branching random walks and the filling scheme in ergodic theory. This is joint work with Julien Barral.
Y. Kifer, Geometric distribution for numbers of returns in open dynamics
A series of papers studied distributions of numbers of returns to shrinking targets for various dynamical systems. Another series of papers stud ied open dynamical systems describing their behavior until exiting through a ”hole”. We combine both setups studying distributions of numbers of returns to a shrinking target before hitting a shrinking ”hole” which under certain con ditions become geometric. Considering ψmixing dynamical systems allows us to deal with a multiple return (nonconventional) setup while restriction our selves to φmixing systems enables us to deal with dynamical systems modeled by some Young towers such as GibbsMarkov ones.
V. Kleptsyn, The Furstenberg theorem : adding a parameter and removing the stationarity.
The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices from SL(n,R); their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices from SL(2,R) depend on an additional parameter. It turns out that in this new situation, the conclusion changes. Namely, under some natural conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes.
Another direction of the generalization of the classical Furstenberg theorem is removing the stationarity assumption. That is, the matrices that are multiplied are still independent, but no longer identically distributed. Though in this setting most of the standard tools are no longer applicable (no more stationary measure, no more Birkhoff ergodic theorem, etc.), it turns out that the Furstenberg theorem can (under the appropriate assumptions) still be generalized to this setting, with a deterministic sequence replacing the Lyapunov exponent.
The two generalizations can be mixed together, providing the Anderson localization conclusions for the nonstationary 1D random Schrödinger operators.
marked length spectrum on closed Riemannian manifolds of negative curvature.
This is joint work with Colin Guillarmou and Thibault Lefeuvre.
Decay of correlations for Axiom A flows remains an open and notoriously difficult problem. A reasonable conjecture is that an open and dense set of Axiom A flows have exponential decay of correlations for all nontrivial hyperbolic basic sets, but despite some isolated positive results this conjecture seems beyond modern technology. A more tractable problem is to prove rapid mixing (mixing at all polynomial orders). This property is known to be open and dense, but only for observables that are sufficiently smooth. In this talk, we explain how to remove this restriction, proving that for an open and dense set of Axiom A flows the nontrivial hyperbolic basic sets are rapid mixing for Holder observables. Consequences and extensions will be mentioned. (Joint work with Caroline Wormell.)
but not surjective. Any map f has a rotation number and can be described in terms of
three parameters: λ, δ, μ. The dynamics of f will be fully described by two functions of
the parameters whose definitions use HeckeMahler series. As an application, we prove
that the rotation number takes a rational value whenever the parameters are algebraic
numbers. This result extends our previous theorem about the case where f is a circle
contracted rotation meaning that the map has constant slope. The talk is based on a joint
work with Michel Laurent.
M. Peigne, On the offine recursion on dimension ≥2 in the critical case.
We fix d ≥ 2 and denote S the semigroup of d × d matrices with non negative entries. We consider a sequence (A_{n}, B_{n}), n≥1, of i. i. d. random variables with values in Sx(R^{+})^{d} and study the asymptotic behavior of the Markov chain (X_{n})_{n≥0 }on (R^{+})^{d} defined by:
∀n ≥ 1, X_{n+1} = A_{n+1}X_{n} + B_{n+1},
where X_{0} is a fixed random variable. We assume that the Lyapunov exponent of the matrices A_{n} equals 0 and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure λ on (R+)^{d} which is invariant for the chain (X_{n})_{n≥0}. The existence of λ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices. Its unicity is a consequence of a general property, called “local contractivity”, highlighted about 20 years ago by M. Babillot, Ph. Bougerol and L. Elie, then S. Brofferio in the case of the affine recursion in dimension 1. This property has been extensively studied for general iterated function systems by M. Peigné et W. Woess; as far as we know, the affine recursion we present here is the first example in dimension ≥ 2 where this weak contractivity phenomenon is described. (Work in progress with S. Brofferio and T.D.C. Pham).
K. Simon, Assouad dimension for the attractor of typical selfsimilar and selfconformal Iterated Function Systems on the line.
The new results are joint with Balázs Bárány, István Kolossváry and Michal Rams.
Recently, considerable attention has been paid to the study of the socalled Assouad dimension, which was introduced in late seventies related to the study of some embedding problems. After giving an introduction to the Assouad dimension, I will state some of the recent results in fractal geometry related to the Assouad dimension of selfsimilar and selfconformal sets. Finally, I state our result related to the Assouad dimension of typical selfsimilar and selfconformal sets on the line.
H. Takahasi, Existence of large deviations rate function for any real quadratic map
Given time series data, we can consider which point is the largest we’ve seen up to a given time: a “record event”. Classical applications are to fastest times etc in sport and the standard models say, for example, that the expected number of record times within the first n observations should scale log n. In this talk I’ll present joint work with Mark Holland on records for observables in dynamical systems (i.e. the largest value some observable \phi takes along the first n iterates of some point). I’ll introduce the related point processes, convergence of which imply a detailed knowledge of the record times and record values.
E. Verbitskiy, On the relation between onesided and twosided Gibbs measure.
I will discuss two questions arising in thermodynamic formalism on lattices Z_+ and Z. In the first part of the talk, I will present the necessary and sufficient conditions for gmeasures to be the DobrushinLanfordRuelle Gibbs states, and in the second part of the talk, I will discuss a similar question in the nontranslation invariant setting. (Joint work with S. Berghout, A. van Enter, and R. Fernandez).
M. Zworski, MorseSmale flows and internal waves
Colin de Verdi\`ere and SaintRaymond have recently elucidated the connection between dynamics of internal waves in (linearized) stratified fluids and certain MorseSmale flows. I will present an alternative approach to the question from joint work with Dyatlov. When viscosity is present analogues of PollicottRuelle resonances also appear in the subject (joint work with Galkowski).
Titles (25 minute talks)
Berezin, Limiting behavior of additive functionals of point processes
Cantrell, Relative growth in hyperbolic groups
Given a hyperbolic group G and infinite index subgroup N, it is known that, as n increases, the function that counts the number of elements of word length n that belong to N, grows strictly slower than the function that counts the number of elements of word length n. In this talk we discuss how to quantify this difference in growth under additional assumptions on N.
Castorrini,Quantitative statistical properties for a class of two dimensional partially hyperbolic systems.
Cioletti, A Central Limit Theorem for Equilibrium States in Noncompact Setting
Colognese, Volume entropy for flat surfaces
We consider an analogue of Manning's volume entropy in the context of flat surfaces. We show that there is also an analogue of Margulis' asymptotic formula.
Crimmins, Stability of Oseledets splittings for random PerronFrobenius operator cocycles
We consider the problem of stability and approximability of Oseledets splittings and Lyapunov exponents for PerronFrobenius operator cocycles associated to random dynamical systems. By developing a random version of the perturbation theory of Gouezel, Keller, and Liverani, we obtain a general framework for solving such stability problems, which is particularly well adapted to applications to random dynamical systems. We apply our theory to random dynamical systems consisting of C^k expanding maps on S^1 (k ≥ 2) and provide conditions for the stability of Lyapunov exponents and Oseledets splitting of the associated PerronFrobenius operator cocycle to (i) uniformly small fiberwise C^{k−1}perturbations to the random dynamics, and (ii) numerical approximation via a Fejer kernel method. A notable addition to our approach is the use of Saks spaces, which provide a unifying framework for many key concepts in the socalled ‘functional analytic’ approach to studying dynamical systems, such as LasotaYorke inequalities and GouezelKellerLiverani perturbation theory.
Doan, Topological classification of random circle homeomorphisms
In this talk, we present a classification of random orientationpreserving circle homeomorphisms up to topological conjugacy of the random dynamical systems generated by their i.i.d. iterations. This is a joint work with J. Lamb, J. Newman and M. Rasmussen.
Fernando, Edgeworth Expansions for (mostly) Hyperbolic Dynamical Systems
Garcia Zelada, Large deviations for Gibbs measures
A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds
Nakano, Emergence via nonexistence of averages
Inspired by a recent work by P. Berger, we introduce the concept of pointwise emergence. This concept provides with a new quantitative perspective into the study of nonexistence of averages for dynamical systems. We show that high pointwise emergence on a large set appears for abundant dynamical systems: There is a dense subset of any Newhouse open set each element of which has superpolynomial pointwise emergence on a positive Lebesgue measure subset of the state space. Furthermore, the full shift has superpolynomial pointwise emergence on a residual subset of the state space. This is joint work with S. Kiriki and T. Soma.
Selley, A selfconsistent dynamical system with multiple absolutely continuous invariant measures.
In this talk we study a class of selfconsistent dynamical systems, selfconsistent in the sense that the discrete time dynamics is different in each step depending
on some current statistics. Motivated by an example of M. Blank, we concentrate on a special case where the dynamics in each step is an expanding betamap. Included in the definition of beta is a nonnegative parameter controlling the strength of selfconsistency. We show a selfconsistent system which has a unique absolutely continuous invariant measure (acim) if the selfconsistency parameter is zero, but at least two in case of any nonzero value of the parameter. With a slight modification, we transform this system into one which produces a phase transitionlike behavior: it has a unique acim for sufficiently small vales of the paramter, and multiple for sufficiently large values. We discuss the stability of the invariant measures by the help of numerics.
Stadlbauer, Quenched and annealed versions of Ruelle’s PerronFrobenius theorem
Su, Random Young towers and quenched limit laws
We obtain a quenched almost sure invariance principle for Random Young
Tower. Applications are some i.i.d. perturbations of some nonuniformly expanding
maps.
Yang, A countable partition for singular flows, and its application on the entropy theory.
In this talk, we will construct a countable partition $\sA$ for flows with hyperbolic singularities by introducing a new crosssection for each singularity. Such partition turns out to have finite metric entropy for every invariant probability measure. We will use this new construction to study the entropy theory for singular flows away from homoclinic tangencies and show that the entropy function varies upper semicontinuously.
Practical Information
 If you are starting from Marseille(s) Airport (Aeroport Marseille Provence) then usually one takes the coach to St. Charles (the train and bus station) in Marseilles. Buses are every 1020 minutes (last bus 00:50) and the journey takes about 30 minutes and costs 10:00 Euros one way (a little more to get the metro+bus to Luminy included). A return costs 16 Euros. The ticket is usually bought at a Kiosk near to the bus. A taxi from the airport to Luminy would cost close to 100 Euros (I guess).
 If you arrive to St. Charles station before 21:00 then there are two stages to getting to Luminy: (a) Go down the long escalator to the metro station and take line M2 in the direction "Sainte Marguerite Dromel" to the stop "Rond Point du Prado" (the 5th stop) and then head up for daylight; (b) You then take a B1 bus from a stop on the opposite side of the road to the park surrounded by railings in the direction of Luminy. Stay on the bus to the terminus  which on weekends and evenings is the gates to the Luminy campus. During the week the terminus is the centre of the campus of Luminy  and very close to the CIRM building. [A very colourful but not very useful real time map of buses is here]
 If you (luckily) arrive to St. Charles station after 21:00 then there is the direct (night) bus 521 to Luminy (at 21:20, 21:55, 22:20, 22:45, 23:20, 23:40, 00:05, 00:40 modulo some white noise), although this stops at the main gate to the campus of Luminy.
Where is my accommodation?
 If you are staying in the CIRM building then head there (it is a large pink building marked 19 in the middle of the campus map). This applies to most people. You can pick up the keys on the ground floor reception (either from a human, or in an envelope near the entrance with your name on it). There are invariably a number of other experienced participants (with nothing better to do) hanging around who can help in case of doubt.
 If you are one of the four people staying at CROUS then head to Batiment B (it is the middle of three large rectangular building showing wear and tear denoted "B" and "102" on the campus map  bottom right of map). You can collect your key from the reception on the ground floor, which is (hopefully) open 24 hours a day. (They may like to see a passport or equivalent).
 If you arranged accommodation of your own elsewhere  then enjoy!
Where can I eat?
 The regular meals at CIRM start on Monday. There is a selfservice breakfast (at least for people staying in the CIRM building) from 7:009:00. Lunches are from 12:30 and dinner from 19:30.
 On Sunday evening there is a cold buffet from 19:3022:30.
 You might prefer to enjoy dinner in colourful Marseilles (and take one of the night buses [mentioned above] back  they go via the Old Port).
 During the week you may find me taking breakfast (Coffee, Orange Juice and Croissant for 1.90 Euros  card payment required) at the pleasant and inexpensive coffee bar in the geometrically elegant new hexagonal library (the hexagonal building denoted 45 on the campus map). It is open 8:0017:30 for snacks and drinks.
 There is also a standard inexpensive cafetaria (1st floor of the building denoted 4 on the campus map) for alternative lunches and a very small shop (on the ground floor) for bottles of drinks, etc.
What can I do when I am not enjoying the lectures?
 The Luminy Campus is located in a unique national park. After lunch one can walk east about 2025 minutes through the woods to arrive to the coast, with spectacular views of cliffs and bays (Calanques). Straight ahead (slightly to the right) there is an easy walk to a viewpoint. There are various other paths of varying difficulty, shown on a map available at CIRM, several leading down to the shore.
 The library is quite a pleasant place (air conditioned) to work, with a surprisingly good selection of books and helpful librarians.
 For mathematical bibliophiles, the SMF (= French Mathematical Society spelt "backwards") sells books in the next building to the library. My favorite deal is that participants can buy books halfprice (and if they buy two they can choose a freebie  providing it is old enough).
 Further afield, one could spend the free afternoon(s) visiting the beautiful town of Aix (bus to St.Charles and then either short train or bus trip to Aix or Cassis).
 On the free afternoon(s) sufficiently tough rugged mathematician can spend several hours hiking along the coast to Cassis (a small resort town as well as the name of an essential ingredient in Kir). There is a bus back  providing you don't miss it.
 The central focus in Marseille(s) is the Old Port. From there one can take short inexpensive trips on public ferrieswhich go a short distance up and down the coast. There is also a ferry which goes out to the Chateau d'If, the setting for the huge (and hugely entertaining) novel "The count of Monte Cristo".
Workshop in July
Schedule
Monday  Tuesday  Wednesday  Thursday  Friday  
9:0010:00  Froyland  Bandtlow  Bissacot  Rousseau  Haydn 
10:0011:00  Bahsoun  Vtynova  GonzalezTokman  Cipriano  Guilietti 
11:3012:30 
Wormell 
Slipantschuk  H. Zhang  Galatolo  Pene 
Lunch  Lunch  Lunch  Lunch  Lunch  
Hu (14:4015:40)  
16:00 16:30  Jiang  Free afternoon  Freitas (14:5016:40)  
16:3017:30  Fan  Gallavotti  Nicol (17:0018:00)  
18:0019:00  Jenkinson  Ruelle  "Pot" (18:0019:30)  
Dinner  Dinner  Dinner  Dinner  
21:0023:00 
Title and Abstracts
W.Bahsoun, "Linear response for random composition of maps"
I will present a result on linear response for random composition of maps. In particular, I will present interesting applications to random continued fractions (GaussRenyi random map) and to random PomeauManneville maps.
O. Bandtlow, "Spectral approximation of transfer operators"
The talk will be concerned with the problem of how to efficiently approximate spectral data of transfer operators for analytic hyperbolic maps. I will focus in particular on how to obtain explicitly computable error bounds.
This talk is motivated by the following question: How do ergodic properties of flows varies with a time change? I will be interested in studying this question in the case of suspension flows over countable Markov shifts.
I will start by introducing the results we requiere on thermodynamics formalisms for countable Markov shifts. I will then describe our topological description of the space of suspension flows according to certain thermodynamic quantities and I will explain the analytic tools we use to construct examples with prescribed thermodynamic behaviour. Finally, I will explain some properties that we can show, for example, that the set of suspension flows defined over the full shift on a countable alphabet having finite entropy is open. This is joint work with Godofredo Iommi from Pontificia Universidad Católica de Chile.
A. Fan, "
$$
Dvoretzky random covering"
Throw random intervals on the circle
$\mathbb{}\mathbb{}\mathbb{}$
. Assume that the centers are independent and follow a probability law
$$
and that the lengths are given positive numbes less than
$\mathrm{}$
. The
$$
Dvoretzy covering problem is to find conditions for the circle to be covered almost surely by the random intervals or for a given compact set to be covered almost surely. When
$$
is the Lebesgue measure, we find the classical Dvoretzky covering problem (1956), which was solved by L. Shepp (1972) for the circle, by J. P. Kahane for compact sets (1987). Many people were interested in and/or had contributed to this problem and related problems (P. Levy, P. Billard, S. Orey, P. Erd\"{o}s, B. Mandelbrot, J. Hawkes, A. H. Fan, J. Barral, J. Wu et al). Kahane's solution is based on B. Mandelbrot's Poisson model and S. Janson 's stopping time technique. Classical Dvoretzky problem of higher dimension remains unsolved (intervals are replaced by balls or cubes etc). With D. Karagulyan, we consider an absolutely continuous probability measure instead of Lebesgue measure. Under weak regularity condition on the density, we can solve the problem. However efforts are needed for singular probability measures. Let us mention that Fan, Schmeling and Troubetzkoy had considered the case that the centers are the orbits of the doubling dynamics according to a (singular) Gibbs measure (the indepedence is lost in this case). Only a weak result was obtained in this case. The multiplicative chaos is one of our useful tools and we base our discussions on Shepp's and Kahane's results by establishing some comparison principles. Open questions will be mentioned.
J. Freitas, "Rare events on fractal landscapes"
We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a fractal set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index (EI). The EI turns out to be very useful to identify the compatibility between the dynamics and the geometrical structure of the fractal set.
G. Froyland, "Fourier approximation of the statistical properties of Anosov maps on tori"
We study the stability of statistical properties of Anosov maps on tori by examining the stability of the spectrum of an analytically twisted PerronFrobenius operator on the anisotropic Banach spaces of Gouëzel and Liverani. We obtain the stability of various statistical properties (the variance of a CLT and the rate function of an LDP) of Anosov maps to general perturbations, including new classes of numerical approximations. In particular, we obtain new results on the stability of the rate function under deterministic perturbations. As a key application, we focus on perturbations arising from numerical schemes and develop two new Fourieranalytic method for efficiently computing approximations of the aforementioned statistical properties. This includes the first example of a rigorous scheme for approximating the peripheral spectral data of the PerronFrobenius operator of an Anosov map without mollification. Using the two schemes we obtain the first rigorous estimates of the variance and rate function for Anosov maps.
S. Galatolo, "Quantitative statistical stability in random systems, computer aided proofs linear response"
Dynamical systems perturbed by noise appear naturally as models of physical and social systems. The presence of noise and its regularizing effects allow a functional analytic approach to be very efficient for the study of the statistical properties of these systems. In several interesting cases this can be approached rigorously by computational methods. As a nontrivial example of this, we prove the existence of noise induced order in the model of chaotic chemical reactions where it was first discovered numerically by Matsumoto and Tsuda in 1983. We show that in this random dynamical system the increase of noise causes the Lyapunov exponent to decrease from positive to negative, stabilizing the system. The method is based on a certified approximation of the stationary measure in the L1 norm. Time permitting we will also talk about linear response of such systems when the deterministic part of the system is perturbed deterministically.
G. Gallavotti, "Nonequilibrium ensembles: NS example''
How to formulate a theory of ensembles analogous to that for the equilibrium ensembles (eg. canonical or microcanoniocal ensembles) todescribe the stationary states out f equilibrium? A proposal is suggested by the example of the NavierStokes equation. The NS equation will be considered for an incompressible fluid in a periodic box and subject to a stirring force constant in time and acting at large scale (ie at the scale of the container). Stationary states depend on a single parameter R=Reynolds number= inverse of viscosity and for a family E of probability distributions on the velocity fields. The possibility of existence of other equations whose stationary states equations whose stationary states have exactly the same distributions through a mechanism analogous to that
for the the equivalence of equilibrium states of different ensembles (which will be proposed to be similar to the equivalence in the thermodynamic limit which in the NS case will correspond to the ultraviolet regularization N >
infinity.
P. Giulietti, "Linear response for dynamical systems with noise"
We present some results on dynamical systems with random noise. By studying an annealed transfer operator, we show that noise allows for linear response under very mild assumptions on the dynamical system. The key argument revolves around controlling the operator by pairs of norms, such as a Wassersteintype norm and L1, which take into account both the longterm behavior of the system and the regularization effect of the noise. Time permitting, I will show how linear response can be proven if assisted by rigorous numeric for some systems which are still out of reach by analytical methods. (joint work with S. Galatolo).
N. Haydn, "Exponential Law for Random Maps on Compact Manifolds"
We consider random dynamical systems on manifolds modeled by a
skew product which have certain geometric properties and whose measures
satisfy quenched decay of correlations at a sufficient rate. We prove
that the limiting distribution for the hitting and return times to
geometric balls are both exponential for almost every realisation. We
then apply this result to random
${}^{\mathrm{}}$
maps of
the interval and random parabolic maps on the unit interval.
This is joint work with J Rousseau and F Yang.
For uniformly hyperbolic systems, it is wellknown that its metric entropy with respect to the SRB measure depends on the system differentiably when the perturbation is sufficiently smooth. We present results on the possible values of the entropy when the system varies. In the Axiom A case, we present the derivative formula for the entropy with respect to the generalized SRB measure and in the dimensiontwo case, the derivative formula for the Hausdorff dimension of the hyperbolic set.
hyperbolic dynamical systems. Joint work with Meagan Carney (Max Planck Institute for Complex Systems Dresden) and Mark Holland (University of Exeter).
J. Rousseau,"On the shortest distance between orbits and the longest common substring problem"
We study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For random processes, this problem corresponds to the longest common substring problem and we will explain how the growth rate of the longest common substring is linked with the Renyi entropy. We will also extend these studies to the realm of random dynamical systems. This includes some joint work with Vanessa Barros and Lingmin Liao and some joint work with Adriana Coutinho and Rodrigo Lambert.
D. Ruelle, "The cascade of eddies in turbulence."
The intermittency exponents in turbulence are well described by a cascade of eddies of size tending to zero, which is also compatible with other turbulence features. Can one give a good probabilistic description of the decay of an eddy into smaller eddies?
J. Slipantschuk, "PollicottRuelle resonances for analytic expanding and hyperbolic maps."
Using analytic properties of Blaschke factors, we construct families of expanding circle maps as well as families of hyperbolic toral diffeomorphismsfor which the spectra of the associated transfer operators acting on suitable Hilbert spaces can be computed explicitly.
P. Vytnova, "Illusions: curves of zeros of Selberg zeta functions"
It is well known (since 1956) that the Selberg Zeta function for compact surfaces satisfies the "Riemann Hypothesis": any zero in the critical strip 0<R(s)<1 is either real or Im(s)=1/2. The question of location and distribution of the zeros of the Selberg Zeta function associated to a noncompact hyperbolic surface attracted attention of the mathematical
community in 2014 when numerical experiments by D. Borthwick showed that for certain surfaces zeros seem to lie on smooth curves. Moreover, theindividual zeros are so close to each other that they give a visual impression that the entire curve is a zero set.
We will give an overview of the computational methods used, present recent results, justifying these observations as well as state open conjectures.
C. Wormell, "Spectral Galerkin transfer operator methods in uniformlyexpanding dynamics"
Fullbranch uniformly expanding maps and their longtime statistical quantities are commonly used as simple models in the study of chaotic dynamics, as well as being of their own mathematical interest. A wide range of algorithms for computing these quantities exist, but they are typically unspecialised to the highorder differentiability of many maps of interest, and consequently have a weak tradeoff between computational effort and accuracy.
This talk will cover a rigorous method to calculate statistics of these maps by discretising transfer operators in a Chebyshev polynomial basis. This discretisation is highly efficient: I will show that, for analytic maps, numerical estimates obtained using this discretisation converge exponentially quickly in the order of the discretisation, for a polynomially growing computational cost. In particular, it is possible to produce (nonvalidated) estimates of most statistical properties accurate to 14 decimal places in a fraction of a second on a personal computer. Some applications of the method to exploration of questions in dynamics will be presented.
Practical Information
 If you are starting from Marseille(s) Airport (Aeroport Marseille Provence) then usually one takes the coach to St. Charles (the train and bus station) in Marseilles. Buses are every 1020 minutes (last bus 00:50) and the journey takes about 30 minutes and costs 8:30 Euros one way (although for 9.20 Euros one can also get the metro+bus to Luminy included). The ticket is usually bought at a Kiosk next to the bus. A taxi from the airport to Luminy would cost close to 100 Euros (I guess). [The more adventurous might fly to Nimes. Montpellier or Nice and then take a bus or train to St. Charles  which is great fun, but takes longer]
 If you arrive to St. Charles station before 21:00 then there are two stages to getting to Luminy: (a) Go down the long escalator to the metro station and take line M2 in the direction "Sainte Marguerite Dromel" to the stop "Rond Point du Prado" (the 5th stop) and then head up for daylight; (b) Youthen take a B1 bus from a stop on the opposite side of the road to the green area surrounded by railings (Velodrome) in the direction of Luminy. Stay on the bus to the terminus  which is the centre of the campus of Luminy  and very close to the CIRM building. [A very colourful but not very useful real time map of buses is here]
 If you (luckily) arrive to St. Charles station after 21:00 then there is the direct (night) bus 521 to Luminy (at 21:20, 21:55, 22:20, 22:45, 23:20, 23:40, 00:05, 00:40 modulo some white noise), although this stops at the main gate to the campus of Luminy.
Where is my accommodation?
 If you are staying in the CIRM building then head there (it is a large pink building marked 19 in the middle of the campus map). This typically applies to people who are staying just for one week and who are not traveling with a partner. You can pick up the keys on the ground floor reception (either from a human, or in an envelope near the entrance with your name on it). There are invariably a number of other experienced participants (with nothing better to do) hanging around who can help in case of doubt.
 If you are staying at CROUS then head to Batiment B (it is the middle of three large rectangular building showing wear and tear denoted "B" and "102" on the campus map  bottom right of map). This typically applies to people who are staying for two weeks or are traveling with a partner.You can collect your key from the reception on the ground floor, which is (hopefully) open 24 hours a day. (They may like to see a passport or equivalent).
 If you arranged accommodation of your own elsewhere  then enjoy!
Where can I eat?
 The regular meals at CIRM start on Monday. There is a selfservice breakfast (at least for people staying in the CIRM building) from 7:009:00. Lunches are from 12:30 and dinner from 19:30.
 On Sunday evening there is a cold buffet (at least for people staying in the CIRM building) from 19:3022:30.
 You might prefer to enjoy dinner in colourful Marseilles (and take one of the night buses [mentioned above] back  they go via the Old Port).
 During the week you may find me taking breakfast (Coffee, Orange Juice and Croissant for 1.90 Euros  card payment required) at the pleasant and inexpensive coffee bar in the geometrically elegant new hexagonal library (the hexagonal building denoted 45 on the campus map). It is open 8:0017:30 for snacks and drinks.
 There is also a standard inexpensive cafetaria (1st floor of the building denoted 4 on the campus map) for alternative lunches and a very small shop (on the ground floor) for bottles of drinks, etc.
What can I do when I am not enjoying the lectures?
 The Luminy Campus is located in a unique national park. After lunch one can walk east about 2025 minutes through the woods to arrive to the coast, with spectacular views of cliffs and bays (Calanques). Straight ahead (slightly to the right) there is an easy walk to a viewpointcampus map. There are various other paths of varying difficulty, shown on a map available at CIRM, several leading down to the shore.
 The library is quite a pleasant place (air conditioned) to work, with a surprisingly good selection of books and helpful librarians.
 For mathematical bibliophiles, the SMF (= French mathematical society spelt "backwards") sells books in the next building to the library. My favorite deal is that participants can buy books halfprice (and if they bcampus mapuy two they can choose a freebie  providing it is old enough, e.g., Asterisque 187188?)
 Further afield, one could spend the free afternoon(s) visiting the beautiful town of Aix (bus to St.Charles and then either short train or bus trip to Aix). There is a famous opera and music festival at this time  which I always enjoy.
 On the free afternoon(s) there are usually a cliche of tough rugged mathematicians who spend several hours hiking along the coast to Cassis (a small resort town as well as the name of an essential ingredient in Kir). There is a bus back  providing you don't miss it.
 The central focus in Marseille(s) is the Old Port. From there one can take short inexpensive trips on public ferries which go a short distance up and down the coast. There is also a ferry which goes out to the Chateau d'If, the setting for the huge (and hugely entertaining) novel "The count of Monte Cristo".
 If one was lucky enough to get a supercheap flight to Nimes (beautiful roman artifacts!) or Montepellier then make the time to look around en route (pardon the accidental use of french!)