# Dynamical Systems, Statistical Properties and their Applications

There will be a conference at CIRM-Luminy, Marseille(s), France, in the general area of

### Dynamical Systems, Statistical properties and their Applications

from 9-13 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.

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 8:50-9:00 Opening 9:00-9:40 Berger Knieper Saussol Jin 9:50-10:30 Takahasi Climenhaga Todd Fraser Nogueira 10:55-11:35 Marmi Fisher Holland Falconer Melbourne 11:40-12:20 Short Talks I Peigne Varandas Short talks IV Buzzi Lunch interval 14:00-15:55 Short talks II Short talks III Short talks V 16:00-16:40 Kifer Bufetov Free afternoon Seuret 16:50-17:30 Kleptsyn Naud Verbitsky 17:55-18:35 Conze Zworski Gouezel 18:45-19:25 Ledrappier Dinner Bouillabaisse UK Election

#### Short talks

 Short talks I Short talks II Short talks III Short talks IV Short talks V Nakano(11:40-12:05) Cantrell(14:00-14:25) Colognese(14:30-14:55) Selley(15:00-15:25) Su(15:30-15:55) GarciaZelada(14:00-14:25) Crimmins(14:30-14:55) Doan(15:00-15:25) Berezin(15:30-15:55) Stadlbauer(11:40-12:05) Fernadno(14:00-14:25) Cioletti(14:30-14:55) Castorrini(15:00-15:25) Yang(15:30-15:55)

##### Titles and Abstracts (40 minute talks)
P. Berger, Emergence of stable components
In a joint work with Sebastien Biebler, we show the existence of a locally dense set of real polynomial automorphisms of $\mathbb C^2$ displaying a stable wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These wandering Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof follows from a real geometrical model which enables us to show the existence of an open and dense set of Cr families of surface diffeomorphisms in the Newhouse domain, each of which displaying a historical, high emergent, wandering domain at a dense set of parameters, for every $2\le r\le \infty$ and $r=\omega$. Hence, this also complements the recent work of Kiriki and Soma, by proving the last Taken's problem in the $C^{\infty}$ and $C^\omega$-case.

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.

J-P. 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 (J-P. 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 ‘Marstrand-type’ theorems on the intermediate dimensions of projections of a set in R^n onto almost all m-dimensional subspaces. This is collaborative work with various combinations of Stuart Burrell, Jonathan Fraser, Tom Kempton and Pablo Shmerkin.

T. Fisher, A dichotomy for measures of maximal entropy
I will discuss recent joint work with Jerome Buzzi and Ali Tahzibi. We show that there are open sets of partially hyperbolic diffeomorphisms arbitrarily close to the time-t map of an Anosov flow for which there is a dichotomy for the measures of maximal entropy; either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy where one has a positive central exponent and the other has a negative central exponent.

J. Fraser, Hölder solutions to the winding problem
Given two homeomorphic subsets of the plane, how regular can a homeomorphism be mapping one set to the other? Here, "regular" will refer to Lipschitz or Hölder conditions. I will discuss a concrete example, known as the "winding problem", where the sets in question are a line segment and a spiral. This problem has applications and origins in conformal welding theory and the modelling of turbulence in fluid dynamics.

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.

M. Holland, On almost sure convergence of maxima for dynamical systems

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 non-uniformly 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/non-degenerate 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 Gibbs-like 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 ψ-mixing dynamical systems allows us to deal with a multiple return (nonconventional) setup while restriction our- selves to φ-mixing systems enables us to deal with dynamical systems modeled by some Young towers such as Gibbs-Markov 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 non-stationary 1D random Schrödinger operators.

G. Knieper, Marked Length spectrum rigidity and the geodesic stretch
In this talk I like to report on recent rigidity results for the
marked length spectrum on closed Riemannian manifolds of negative curvature.
This is joint work with Colin Guillarmou and Thibault Lefeuvre.

C. Matheus, Codimension one blenders via renormalization and Marstrand's theorem

I. Melbourne, Uniformly hyperbolic flows: Rapid mixing for Holder observables
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.)

F. Naud, Resonances and spectral gap of random hyperbolic surfaces
After a necessary short survey on the spectral theory of convex co-compact hyperbolic surfaces,
we will define a notion of random covers modelled over certain random regular graphs. In the large degree limit,
we will state some a.a.s explicit spectral gap results that generalise in some sense the famous Selberg 3/16 theorem.

A. Nogueira, Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series
Let f = ffλ,δ,μ be a family of 2-interval affine increasing maps which is injective
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 Hecke-Mahler 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 semi-group of d × d matrices with non negative entries. We consider a sequence (An, Bn), n≥1, of i. i. d. random variables with values in Sx(R+)d and study the asymptotic behavior of the Markov chain (Xn)n≥0 on (R+)d defined by:

∀n ≥ 1, Xn+1 = An+1Xn + Bn+1,

where X0 is a fixed random variable. We assume that the Lyapunov exponent of the matrices An 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 (Xn)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).

S. Seuret, Function spaces in multifractal environment, and the Frisch-Parisi conjecture
Multifractal properties of data coming from many scientific fields (especially in turbulence) are now rigorously established. Unfortunately, the parameters measured on these data do not correspond to those mathematically obtained for the typical (or almost sure) functions in the standard functional spaces: Hölder, Sobolev, Besov…
In this talk, we introduce very natural Besov spaces in which typical functions possess very rich scaling properties, mimicking those observed on data for instance. We obtain various characterizations of these function spaces, in terms of oscillations or wavelet coefficients.
Combining this with the construction of almost-doubling measures with prescribed scaling properties, we are able to bring a solution to the so-called Frisch-Parisi conjecture. This is a joint work with Julien Barral (Université Paris-Nord)

K. Simon, Assouad dimension for the attractor of typical self-similar and self-conformal 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 so-called 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 self-similar and self-conformal sets. Finally, I state our result related to the Assouad dimension of typical self-similar and self-conformal sets on the line.

H. Takahasi, Existence of large deviations rate function for any real quadratic map
We show the level-2 large deviation principle for any real quadratic map.
Our argument can be generalized to any S-unimodal map with non-flat critical point.
This is a joint work with Masato Tsujii (Kyushu University).

M.Todd, Record events in dynamical systems
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.

P. Varandas, Thermodynamic formalism for some non-uniformly expanding maps
We discuss thermodynamic formalism for classes of random maps exhibiting non-uniformly expanding behavior. In brief terms, these are such that the random dynamical system is driven by an invertible ergodic transformation and so that expanding and non-expanding behaviors may coexist for all generating
dynamics (no Markov assumption is required). Under mild combinatorial and transitivity assumptions
we prove that equilibrium states associated to hyperbolic potentials exist, are unique, and have exponential
decay of correlations. This is a joint work with Manuel Stadlbauer (Federal University of Rio de Janeiro) and
Shintaro Suzuki (Keio University).

E. Verbitskiy, On the relation between one-sided and two-sided 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 g-measures to be the Dobrushin-Lanford-Ruelle Gibbs states, and in the second part of the talk, I will discuss a similar question in the non-translation invariant setting. (Joint work with S. Berghout, A. van Enter, and R. Fernandez).

M. Zworski, Morse--Smale flows and internal waves
Colin de Verdi\ere and Saint-Raymond have recently elucidated the connection between dynamics of internal waves in (linearized) stratified fluids and certain Morse--Smale flows. I will present an alternative approach to the question from joint work with Dyatlov. When viscosity is present analogues of Pollicott--Ruelle 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 Non-compact 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 Perron--Frobenius operator cocycles

We consider the problem of stability and approximability of Oseledets splittings and Lyapunov exponents for Perron-Frobenius 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 Perron-Frobenius operator cocycle to (i) uniformly small fiber-wise 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 so-called ‘functional analytic’ approach to studying dynamical systems, such as Lasota-Yorke inequalities and Gouezel-Keller-Liverani perturbation theory.

Doan, Topological classification of random circle homeomorphisms

In this talk, we present a classification of random orientation-preserving 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 non-existence 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 non-existence 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 super-polynomial pointwise emergence on a positive Lebesgue measure subset of the state space. Furthermore, the full shift has super-polynomial pointwise emergence on a residual subset of the state space. This is joint work with S. Kiriki and T. Soma.

Selley, A self-consistent dynamical system with multiple absolutely continuous invariant measures.

In this talk we study a class of self-consistent dynamical systems, self-consistent 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 beta-map. Included in the definition of beta is a nonnegative parameter controlling the strength of self-consistency. We show a self-consistent system which has a unique absolutely continuous invariant measure (acim) if the self-consistency 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 transition-like 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 Perron-Frobenius 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 non-uniformly 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 cross-section 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 semi-continuously.

##### Practical Information
How do I get to CIRM-Luminy? (Unofficial Guide - the official guide is here)
• 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 10-20 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 self-service breakfast (at least for people staying in the CIRM building) from 7:00-9:00. Lunches are from 12:30 and dinner from 19:30.
• On Sunday evening there is a cold buffet from 19:30-22:30.
• 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 8:00-17:30 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 20-25 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 half-price (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 9:00-10:00 Froyland Bandtlow Bissacot Rousseau Haydn 10:00-11:00 Bahsoun Vtynova Gonzalez-Tokman Cipriano Guilietti 11:30-12:30 Wormell Slipantschuk H. Zhang Galatolo Pene Lunch Lunch Lunch Lunch Lunch Hu (14:40-15:40) 16:00 -16:30 Jiang Free afternoon Freitas (14:50-16:40) 16:30-17:30 Fan Gallavotti Nicol (17:00-18:00) 18:00-19:00 Jenkinson Ruelle "Pot" (18:00-19:30) Dinner Dinner Dinner Dinner 21:00-23:00

#### 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 (Gauss-Renyi random map) and to random Pomeau-Manneville 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.

R. Bissacot Thermodynamic Formalism on Generalized Symbolic Spaces''
In 1999 R. Exel and M. Laca extended the construction of the Cuntz-Krieger algebras for infinite countable symbols and transitive matrices, they introduced a class of algebras which today are known as Exel-Laca algebras $O_A$, which are related to its respective countable Markov shift $\Sigma_A$, similarly with what happens to the Cuntz-Krieger algebras. Such construction gave birth a locally compact version of the standard symbolic space $\Sigma_A$, the space $X_A$, which in general contains the usual space $\Sigma_A$ as a dense subset and, when $\Sigma_A$ is locally compact these two spaces coincide. On another hand, M. Denker, M. Urbański, O. Sarig, and many others developed the Thermodynamic Formalism for the standard countable Markov shifts $\Sigma_A$, these spaces, in general, are not locally compact. Despite a big success of Exel-Laca algebras in the Operator Algebra community, the measure-theoretic aspects of this generalization of the symbolic space and interaction with the dynamical system community were essentially zero until now. We obtained the first results about thermodynamic formalism (conformal and DLR measures, pressure, phase transitions, etc.) on the space $X_A$. We will give some geometric interpretation of $X_A$, results about the existence of conformal measures on this new setting and we will answer the first natural question: If $\Sigma_A$ is a subset of $X_A$, what is the connection between the standard thermodynamic formalism on $\Sigma_A$ and the results on $X_A$? We will see that not only new phenomena appear (phase transitions) as well we can recover conformal measures living on $\Sigma_A$ as a limit of new conformal measures which are detected only in the new space $X_A$. The results are part of a project which is still in developing with T. Razseja (IME-USP), Ruy Exel (UFSC) and Rodrigo Frausino (IME-USP).
I. Cipriano, "Time change for flows"

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  . 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 Perron-Frobenius 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 Fourier-analytic 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 Perron-Frobenius 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, "Non-equilibrium 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 Navier-Stokes 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 Wasserstein-type norm and L1, which take into account both the long-term 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).

C. Gonzalez-Tokman, "A spectral approach for quenched limit theorems for random dynamical system"
Random or non-autonomous dynamical systems provide very flexible models for the study of forced or time-dependent systems, with driving mechanisms allowed to range from deterministic forcing to stationary noise. In this talk, we present a spectral approach to the study of non-autonomous dynamics, developed in the last decade, using multiplicative ergodic theory. We then show how spectral methods can be used to establish quenched limit theorems for a class of non-autonomous dynamical systems. (Joint work with Davor Dragicevic, Gary Froyland and Sandro Vaienti.)

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 ${}^{}$ maps of
the interval and random parabolic maps on the unit interval.
This is joint work with J Rousseau and F Yang.

Huyi Hu, "The essential coexistence phenomenon in Hamiltonian dynamics"
We construct an example of a Hamiltonian flow ${}^{}$ on a 4-dimensional smooth manifold  which after being restricted to an energy surface ${}_{}$ demonstrates essential coexistence of regular and chaotic dynamics, that is, there is an open and dense ${}^{}$ -invariant subset  of ${}_{}$ such that restricted to  ${}^{}$ has non-zero Lyapunov exponents in all directions, except the direction of the flow, and is a Bernoulli flow while on the boundary of  , which has positive volume, all Lyapunov exponents of the system are zero. This is a joint work with Jianyu Chen, Yakov Pesin and Ke Zhang

O. Jenkinson, "Validated numerics for transfer operators and their determinants"
M. Jiang, "Variation of Entropy of SRB measures of Uniformly Hyperbolic Systems"
For uniformly hyperbolic systems, it is well-known 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 dimension-two case, the derivative formula for the Hausdorff dimension of the hyperbolic set.
M. Nicol, "Extremes for energy-like observables on hyperbolic systems"
Consider an ergodic measure preserving dynamical system  , and a observable $\mathbb{}$ . We establish limit laws for the maximum process ${}_{}\underset{}{}{}^{}$ in the case where  is maximized on a curve or invariant subspace (rather than at a unique point in phase space) for certain classes of
hyperbolic dynamical systems. Joint work with Meagan Carney (Max Planck Institute for Complex Systems Dresden) and Mark Holland (University of Exeter).
F.Pene, "Functional limit theorems for semi-dispersing billiards with cusps"
We study stochastic properties of billiards in compact domains with convex scatterers and cusps. More precisely, we are interested in the asymptotic behaviour of ergodic sums of Hölder continuous functions. For cusps with ordinary contact, a functional limit theorem with a non standard normalization has been proved by Bálint, Chernov and Dolgopyat. For billiards with a single symmetric cusp of higher flatness, Jung and Zhang proved a non standard limit theorem (convergence to a stable random variable). We extend this result by proving a non standard functional limit theorem (convergence to a Lévy process) for more general billiards with cusps (allowing several cusps, with more general shape, possibly assymetric, with possibly different flatness). This is a joint work with Paul Jung and Hong-Kun Zhang.

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, "Pollicott-Ruelle 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 uniformly-expanding dynamics"
Full-branch uniformly expanding maps and their long-time 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 high-order 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 (non-validated) 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.

H. Zhang, “Length spectrum rigidity for Bunimovich Stadia
We establish the dynamical spectral rigidity for piecewise analytic Bunimovich stadia and squash-type stadia. In addition, for smooth Bunimovich stadia and squash-type stadia we compute the Lyapunov eigenvalues along the maximal period two orbit, as well as value of the Peierls’ Barrier function from the marked length spectrum. This is a joint work with Jianyu Chen and Vadim Kaloshin.

##### Practical Information
How do I get to CIRM-Luminy? (Unofficial Guide - the official guide is here)
• 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 10-20 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 self-service breakfast (at least for people staying in the CIRM building) from 7:00-9:00. 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 19:30-22:30.
• 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 8:00-17:30 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 20-25 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 half-price (and if they bcampus mapuy two they can choose a freebie - providing it is old enough, e.g., Asterisque 187-188?)
• 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 super-cheap flight to Nimes (beautiful roman artifacts!) or Montepellier then make the time to look around en route (pardon the accidental use of french!)