Related papers: Fourier transforms of Gibbs measures for the Gauss…

We generalize an equidistribution theorem \`a la Bader-Muchnik for operator-valued measures constructed from a family of boundary representations associated with Gibbs measures in the context of convex cocompact discrete group of isometries…

Let $\mu$ be a Borel probability measure on $\mathrm{SL}_2(\mathbb R)$ with a finite exponential moment, and assume that the subgroup $\Gamma_{\mu}$ generated by the support of $\mu$ is Zariski dense. Let $\nu$ be the unique…

In this paper we define distance expanding random dynamical systems. We develop the appropriate thermodynamic formalism of such systems. We obtain in particular the existence and uniqueness of invariant Gibbs states, the appropriate…

We investigate ergodic-theoretical quantities and large deviation properties of one-dimensional intermittent maps, that have not only an indifferent fixed point but also a singular structure such that the uniform measure is invariant under…

Let $\Phi$ be a $C^{1+\gamma}$ smooth IFS on $\mathbb{R}$, where $\gamma>0$. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points $x$ that are absolutely normal. That is,…

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an…

A breakthrough result of B\'ar\'any, Hochman and Rapaport published in 2019 established that every self-affine measure on $\mathbb{R}^2$ satisfying certain mild non-degeneracy conditions has Hausdorff dimension equal to its Lyapunov…

Let $\pi:X\to Y$ be a factor map, where $(X,\sigma_X)$ and $(Y,\sigma_Y)$ are subshifts over finite alphabets. Assume that $X$ satisfies weak specification. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\geq 0$. Let $f$ be a continuous…

Let $\nu$ be the Furstenberg measure associated with a non-elementary probability measure $\mu$ on SL_2(R). We show that, when $\mu$ has a finite second moment, the Fourier coefficients of $\nu$ tend to zero at infinity. In other words,…

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ \mu_\omega = \mathop{\circledast}_{k=1}^{\infty} \left( \frac{\delta_0 + \delta_{\lambda_1 \lambda_2 \ldots \lambda_{k-1}…

We prove that the pushforwards of a very general class of fractal measures $\mu$ on $\mathbb{R}^d$ under a large family of non-linear maps $F \colon \mathbb{R}^d \to \mathbb{R}$ exhibit polynomial Fourier decay: there exist $C,\eta>0$ such…

In this article we study the generalized Fourier dimension of the set of Liouville numbers $\mathbb{L}$. Being a set of zero Hausdorff dimension, the analysis has to be done at the level of functions with a slow decay at infinity acting as…

We prove the following. Let $\mu_{1},\ldots,\mu_{n}$ be Borel probability measures on $[-1,1]$ such that $\mu_{j}$ has finite $s_j$-energy for certain indices $s_{j} \in (0,1]$ with $s_{1} + \ldots + s_{n} > 1$. Then, the multiplicative…

We prove that any stable method for resolving the Gibbs phenomenon - that is, recovering high-order accuracy from the first $m$ Fourier coefficients of an analytic and nonperiodic function - can converge at best root-exponentially fast in…

We construct an invariant measure $\mu$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of $\mu$ are initial conditions of global, unique solutions of SQG, that depend continuously on the…

We study how physical measures vary with the underlying dynamics in the open class of $C^r$, $r>1$, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs $u$-state is positive. If transitive,…

Minkowski's question mark function is strictly increasing on $[0, 1]$ and hence defines a Stieltjes measure on $[0, 1]$. A problem originating from Salem in 1943, is to determine whether the Fourier series of this measure decay to zero or…

We study the Hausdorff dimension of Gibbs measures with infinite entropy with respect to maps of the interval with countably many branches. We show that under simple conditions, such measures are symbolic-exact dimensional, and provide an…

We study probabilistic iterated function systems (IFS), consisting of a finite or infinite number of average-contracting bi-Lipschitz maps on R^d. If our strong open set condition is also satisfied, we show that both upper and lower bounds…

Given a finite-to-one map acting on a compact metric space, one classically constructs for each potential in an appropriate Banach space of functionsa transfer operator acting on functions. Under suitable condition, the…