Abstract : With A. Gogolev, we recently found a way to prove the smoothness of conjugacy via transfer operator methods (Perron-Frobenius). So, my idea is to first give an overview of results, old and new, and present one of the driving questions: (Khalil-Lafont) 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?

The question needs to be stated a little bit more carefully to avoid obvious counterexamples.

After this general discussion, the idea is to show a complete answer to the analogous question for expanding maps in any dimension.

And the reminding of the lectures will be to build the setting that will make the proof work also for Anosov flows with foliations.

In the setting of Anosov flow, some sharpening of Livisic results will also be presented, much along the lines of the notion of homologically full Anosov flows. For instance the following statement: Given a homologically full Anosov vector field X, if the integral of a smooth function u vanishes along every homologically trivial orbit, then there is a closed 1-form w such that

C. Liverani, "Averaging and beyond for fast-slow hyperbolic systems"

F. Faure and M. Tsujii, "Micro-local methods in hyperbolic dynamics"

In hyperbolic dynamics (Anosov) each trajectory is strongly unstable and its behavior is unpredictable. A smooth probability distribution evolves also in a complicated way since it acquires higher and higher oscillations. Nevertheless using micro-local analysis, this evolution is predictable in the sense of distributions. It is similar to a quantum scattering problem in cotangent space. In this series of lectures we will use wave-packet (or wavelet) transform and explain how to derive some spectral properties of the dynamics that describe this predictable behavior, as the existence of the intrinsic discrete spectrum of Ruelle-Pollicott resonances, a fractal Weyl law, estimates on the wave front set of the resonances, and band structure of the spectrum in the case of a geodesic flow. This spectrum governs the decay of correlation expansions and may be used to deduce statistical properties of the flow.
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.

There will also be number of keynote lectures, including:

When a dynamical system admitting a natural (SRB) measure is perturbed, it is natural to ask how the SRB measure responds to the perturbation. In the tamest cases, this response is linear, and the derivative of the SRB measure with respect to the parameter can be expressed as a sum of decorrelations (involving the derivative of the system with respect to the parameter). In more subtle situations - for example, systems with bifurcations, or observables with singularities - the SRB measure may be a Hölder function of the parameter. This talk will present a panorama of results about linear and fractional response.

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.

I will present some joint works with Volker Mayer in which we primarily show that for a large class of entire and meromorphic transcendental functions the full geometric thermodynamic formalism holds. Most notably, this means that the transfer operators generated by geometric potentials are well dened and bounded after an appropriate conformal change of Riemannian metric on the complex plane C. We show that these operators are quasi-compact of diagonal type with one leading eigenvalue, which
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.

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