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.[-]

We consider quasi-compact linear operator cocycles driven by an invertible ergodic process and small perturbations of this cocycle. We prove an abstract pathwise first-order formula for the leading Lyapunov multipliers. This result does not rely on random driving and applies also to sequential dynamics. We then consider the situation where the linear operator cocycle is a weighted transfer operator cocycle induced by a random map cocycle. The perturbed transfer operators are defined by the introduction of small random holes, creating a random open dynamical system. We obtain a first-order perturbation formula for the Lyapunov multipliers in this setting. Our new machinery is then deployed to create a spectral approach for a quenched extreme value theory that considers random dynamics with general ergodic invertible driving, and random observations. Further, in the setting of random piecewise expanding interval maps, we establish the existence of random equilibrium states and conditionally invariant measures for random open systems via a random perturbative approach. Finally we prove quenched statistical limit theorems for random equilibrium states arising from contracting potentials. We will illustrate the theory with some explicit examples.[-]

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).[-]

The SRB measure of Sinai billiard maps and flows has been studied for decades, but other equilibrium states have been investigated only recently. Assuming finite horizon, the measure of maximal entropy (MME) of the (discontinuous) map has been constructed and shown to be unique and Bernoulli (joint work with Demers, 2020), under a mild condition (believed to be generic) on the topological entropy. Demers and Korepanov have recently shown that this MME mixes at least polynomially (for H¨older observables). In spite of the continuity of the billiard flow, the mere existence of the MME for the flow has been a challenging problem. I will explain how we obtain existence, uniqueness and Bernoullicity of the MME of the Sinai billiard flow, assuming finite horizon and a mild condition (also believed to be generic), by bootstrapping on very recent work of J´erˆome Carrand about a family of equilibrium states for the billiard map. We use transfer operators acting on anisotropic Banach spaces. (Joint work with J´erˆome Carrand and Mark Demers).[-]

An early motivation of smooth ergodic theory was to provide a mathematical account for the unpredictable, chaotic behavior of real-world fluids. While many interesting questions remain, in the last 25 years significant progress has been achieved in understanding models of fluid mechanics, e.g., the Navier-Stokes equations, in the presence of stochastic driving. Noise is natural for modeling purposes, and certain kinds of noise have a regularizing effect on asymptotic statistics. These kinds of noise provide an effective technical tool for rendering tractable otherwise inaccessible results on chaotic regimes, e.g., positivity of Lyapunov exponents and the presence of a strange attractor supporting a physical (SRB) measure. In this talk I will describe some of my work in this vein, including a recent result with Jacob Bedrossian and Sam Punshon-Smith providing positive Lyapunov exponents for f inite-dimensional (a.k.a. Galerkin) truncations of the Navier-Stokes equations.[-]

We consider stochastic models of scalable biological reaction networks in the form of continuous time pure jump Markov processes. The study of the mean field behavior of such Markov processes is a classical topic, with fundamental results going back to Kurtz, Athreya, Ney, Pemantle, etc. However, there are still questions that are not completely settled even in the case of linear reaction rates. We study two such questions. First is to characterize all possible rescaled limits for linear reaction networks. We show that there are three possibilities: a deterministic limit point, a random limit point and a random limit torus. Second is to study the mean field behavior upon the depletion of one of the materials. This is a joint work with Lai-Sang Young.[-]