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[-]

$Let (X,T)$ 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 separately 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.[-]

We study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted regular periodic orbits are equidistributed relative to these unique equilibrium states.[-]

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.[-]

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.[-]

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 $C^{r}$ 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\leq r\leq \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.[-]

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.[-]