# Summer School in CIRM-Luminy (Marseille) from 1-5 July

**Provisional timetable**

FRH=Federico Rodrigues Hertz

F/T = Faure+Tsujii

FP=Paulin

CL=Liverani

Monday | Tuesday | Wednesday | Thursday | Friday | |

9:00-10:00 | FRH | FRH | CL | FRH | FRH |

10:00-11:00 | F/T | F/T | F/T | F/T | F/T |

11:30-12:30 | FP | FP | FP | FP | CL |

Lunch | Lunch | Lunch | Lunch | Lunch | |

2:30-4:00 | Short talks | Posters | Short talks | Departure | |

4:00-5:00 | Sharp | CL | Free afternoon | CL | |

5:30-6:30 | Kao | Chazottes | Baladi | ||

6:30-7:30 | Buzzi | Urbanski | Lucarini | ||

Dinner | Dinner | Dinner | Dinner | ||

9:00-11:00 | Evening Activities | Evening Activities | Evening Activities | Evening Activities |

##### Titles and Abstracts

There will be four main lecture courses:

**F. Paulin** *``Rate of mixing for geodesic flows with Gibbs measures in negative curvature and trees''*

Abstract : Chaotic dynamical systems as geodesic flows in negative curvature or shifts in symbolic dynamics often have strong mixing properties. This series of four lectures, we will start by giving an explicit construction a la Patterson-Sullivan-Bowen-

**F. Rodriguez Hertz ***"Introduction to rigidity in hyperbolic dynamics"*

*Given two metrics of negative sectional curvature on a manifold and 2 functions with same integral along corresponding closed geodesics, are the metrics isometric and the functions matching?*

*w*such that

**C. Liverani**,

*"Averaging and beyond for fast-slow hyperbolic systems"*

**F. Faure**and**M.****Tsujii**, "Micro-local methods in hyperbolic dynamics"Part 1: for an Anosov flow, we present a semi-classical approach using wave packet transform, giving the discrete spectrum and fractal Weyl law.

Part 2: for an Anosov geodesic flow, we explain more specifically, the band spectrum and the semi-classical zeta function.

**V. Baladi**, "Linear and fractional response: a survey"**J. Buzzi**, "

*Uniqueness of hyperbolic equilibrium measures in homoclinic classes*"

Works by Sarig and Benovadia have built symbolic dynamics for arbitrary diffeomorphisms of compact manifolds. This shows thatthere can be at most countably many ergodic hyperbolic equilibriummeasures for any Holder continuous or geometric potentials. We will explain how this yields uniqueness inside each homoclinic class of measures, i.e., of ergodic and hyperbolic measures that are homoclinically related. In some cases, further topological or geometric arguments can show global uniqueness. This is a joint work with Sylvain Crovisier and Omri Sarig.

**J.-R. Chazottes**, "

*An introduction to concentration inequalities*"

Let $(X,T)$ be a dynamical system preserving a probability measure $\mu$. A concentration inequality quantifies how small is the probability for

$F(x,Tx,\ldots,T^{n-1}x)$ to deviate from $\int F(x,Tx,\ldots,T^{n-1}x) \mathrm{d}\mu(x)$ by an given amount $u$, where $F:X^n\to\mathbb{R}$ is supposed to be separatelIt is well-known that for compact uniformly hyperbolic systems H\"{o}lder potentials have unique equilibrium states. However, it is much less known for non-uniformly hyperbolic systems. In his seminal work, Knieper proved the uniqueness of the measure of maximal entropy for the geodesic flow on compact rank 1 non-positively curved manifolds. A recent breakthrough made by Burns, Climenhaga, Fisher, and Thompson which extended Knieper's result and showed the uniqueness of the equilibrium states for a large class of non-zero potentials, for instance, H\"{o}lder potentials without carrying full pressure on the singular set. In this talk, I will discuss a further generalization of these uniqueness results, following the scheme of Burns, Climenhaga, Fisher, and Thompson's work, to the setting of geodesic flows over compact rank 1 manifolds without focal points. This work is joint with Dong Chen, Kiho Park.y Lipschitz. The bound on that probability involves a constant $C$ depending only on the dynamical system (thus independent of $n$), and $\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2$. In the best situation, the bound is $\exp(-C u^2/\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2)$.

After explaining how to get such a bound for independent random variables, I will show how to prove it for a Gibbs measure on a shift of finite type with a Lipschitz potential, and present examples of functions $F$ to which one can apply the inequality. Finally, I will survey some results obtained for nonuniformly hyperbolic systems modeled by Young towers.

**L.-Y. Kao**, *"Uniqueness of equilibrium states for geodesic flow over manifolds without focal point"
*It is well-known that for compact uniformly hyperbolic systems H\"{o}lder potentials have unique equilibrium states. However, it is much less known for non-uniformly hyperbolic systems. In his seminal work, Knieper proved the uniqueness of the measure of maximal entropy for the geodesic flow on compact rank 1 non-positively curved manifolds. A recent breakthrough made by Burns, Climenhaga, Fisher, and Thompson which extended Knieper's result and showed the uniqueness of the equilibrium states for a large class of non-zero potentials, for instance, H\"{o}lder potentials without carrying full pressure on the singular set. In this talk, I will discuss a further generalization of these uniqueness results, following the scheme of Burns, Climenhaga, Fisher, and Thompson's work, to the setting of geodesic flows over compact rank 1 manifolds without focal points. This work is joint with Dong Chen, Kiho Park.

* V. Lucarini* (joint work with T, Bodai)

*"Global Stability Properties of the Climate: Melancholia States, Invariant Measures, and Phase Transitions"*

For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in past our planet flipped between these two states. The main physical mechanism responsible for such instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attractions. In the weak noise limit, large deviation laws define the invariant measure and the statistics of escape times. By empirically constructing the instantons, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. At this regard, we relate our results to the debate around the prominence of contigency vs. convergence in biological evolution. Finally, we propose a new method for constructing Melancholia states from direct numerical simulations, thus bypassing the need to use the edge-tracking algorithm

Refs:

Lucarini and Bodai, Phys. Rev. Lett. 122, 158701(2019)

Lucarini and Bodai, arXiv:1903.08348 (2019)

**R.Sharp,**"

*Entropy and growth of periodic orbits for Anosov flows and their covers*"

In this talk, we will discuss various growth rates associated to Anosov flows and their covers. The topological entropy of an Anosov flow on a compact manifold is realised as the exponential growth rate of its periodic orbits. If we pass to a regular cover of the manifold then we can consider a corresponding growth rate for the lifted flow. This growth is bounded above by the topological entropy but if the cover is infinite then the growth rate may be strictly smaller. For abelian covers, this phenomenon admits a precise description in terms of a variational principle. More recent work, joint with Rhiannon Dougall, considers more general infinite covers.

**M. Urbanksi**, "Thermodynamic Formalism in Transcendental Dynamics"in addition is simple. In particular, the dual operators have positive eigenvalues and eigenvectors that are Borel probability eigenmeasures. The probability measure obtained by integrating these eigenmeasures against leading eigenfanctions of transfer operators are invariant. We show that these measures are equilibrium states of geometric potentials. The primary applications of these theorems capture the stochastic laws such as exponential decay of correlations, the central limit theorem, and the law of iterated logarithm. it also permits us to provide exact formulas (of Bowen's type) for Hausdorff dimension of

radial Julia sets and multifractal analysis. We will discuss two distinct routes (leading to different though overlapping classes of

meromorphic transcendental functions) to get the geometric thermodynamic formalism. One of them is based on Nevanlina's theory and the other on analogues of integral means spectrum from classical complex analysis of conformal maps.

There will also be some short(er) talks:

**K. Akurugodage. ***Higher order asymptotics for Large Deviations*For sequences of weakly dependent random variables, we obtain asymptotics of all orders for the Large Deviation Principle in the form of an asymptotic expansion. We apply our results to many examples including ergodic sums of smooth expanding maps & subshifts of finite type. In addition, we obtain similar expansions for stochastic processes, and establish them for additive functionals of processes generated from SDEs satisfying the Hörmander condition. This is joint work with Pratima Hebbar.

**C. Gonzalez-Tokman**, *Characterization and perturbations of the Lyapunov spectrum of a class of Perron-Frobenius operator cocycles***
**The Lyapunov spectrum of Perron-Frobenius operator cocycles contains relevant information about dynamical properties of time-dependent (non-autonomous, random) dynamical systems. In this talk we characterize the Lyapunov spectrum of a class of analytic expanding maps of the circle, and discuss stability and instability properties of this spectrum under perturbations. (Joint work with Anthony Quas.)

**J. Leppanen**,* Quasistatic dynamics with intermittency
*Quasistatic dynamical systems (QDS), introduced by Dobbs and Stenlund, model dynamics that transform slowly over time due to external influences. They are generalizations of conventional dynamical systems and belong to the realm of deterministic non-equilibrium processes. The main focus of this talk will be on a particular class of QDSs where the time-evolution is specified by intermittent maps with time-dependent parameters. After defining the model I will present results on its long-term statistical behavior, including a functional central limit theorem.

**R. Moore**, *Properties of the Birhhoff Spectra for the generic continuous functions on a shift space*

**Dmitry Zubov** *Deviation of the averages over the unstable leaves of Anosov diffeomorphisms*

For a C^3 smooth topologically mixing Anosov diffeomorphism with oriented invariant foliations, we show the qualitative equidistribution theorem for the averages of C^2 functions over the (iterated) unstable balls. The key tool is the analysis of the spectrum of the pullback operator acting on a Gouezel-Liverani type Banach space. We show that the eigenfunctions with eigenvalues close to the spectral radius give rise to the families of holonomy invariant finitely additive measures (in the sense of Bufetov and Bufetov-Forni) on the unstable leaves; these finitely additive measures will be shown to control the asymptotics of the considered leafwise integrals.

**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. [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) You take a bus 21Jet or B1 from a stop on the opposite side of the road to the large building 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, 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 (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).

**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 student cafetaria (at the northern end of the building denoted 40 on the campus map)
- There is also an interesting and inexpensive coffee bar in the geometrically elegant new hexagonal library (the hexahttps://www.tripadvisor.fr/Restaurant_Review-g187253-d5256786-Reviews-Les_Terrasses_du_Phoceen-Marseille_Bouches_du_Rhone_Provence_Alpes_Cote_d_Azur.htmlgonal building denoted 45 on the campus map) and a standard cafetaria and small shop across the road from the student cafetaria [mentioned above].
- At the weekend there is no food on Campus between Saturday breakfast and Sunday dinner. The best option is to take the B1 bus to Marseille and to eat there. The entrance to the campus boasts a snack bar (Le luminyen) and a sandwich truck. If you walk down the hill to the first roundabout and then turn left (after about 20 minutes walking) the Tennis Club Phoceen has a fairly pricey buffet lunch every day.

**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 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 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, 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!)