Related papers: Spectrum of weighted Birkhoff average

We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1]^{d}$. Let $E_f({h}) $ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the…

We study pointwise convergence of entangled averages of the form \[ \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f, \] where $f\in…

We study lower and upper bounds of the Hausdorff dimension for sets which are wiggly at scales of positive density. The main technical ingredient is a construction, for every continuum K, of a Borel probabilistic measure \mu with the…

We consider interval maps with countably many full branches and observables with polynomial tails. We show that the Birkhoff spectrum is real analytic and that its convergence to the Hausdorff dimension of the repeller is governed by the…

In this paper we consider the packing spectra for local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the…

We present theoretical calculations of the ensemble-averaged (a.k.a. effective or coherent) wavefield propagating in a heterogeneous medium considered as one realization of a random process. In the literature, it is usually assumed that…

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series admitting meromorphic continuation to the complex plane. Assume we know the location of the poles of $A(s)$ with $|\Im s| \leq T$, and their residues, for some large constant $T$. It is…

Let $S$ be a numerical semigroup of embedding dimension $e$ and conductor $c$. The question of Wilf is, if $\#(\mathbb N\setminus S)/c\leq e-1/e$. \noindent In (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1…

For an infinitely renormalizable negative Schwarzian unimodal map $f$ with non-flat critical point, we analyze statistical properties of periodic points as the periods tend to infinity. Introducing a weight function $\varphi$ which is a…

Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} These averages satisfy a uniform scale-free $ \ell…

We show that on a $\sigma$-finite measure preserving system $X = (X,\nu, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$,…

We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$, also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces $H^{s}_{0}(\mathbb{T},\mathbb{R})$, $s > -1/2$,…

The classical Birkhoff ergodic theorem states that for an ergodic Markov process the limiting behaviour of the time average of a function (having finite $p$-th moment, $p\ge1$, with respect to the invariant measure) along the trajectories…

Consider a H\"older continuous potential $\phi$ defined on the full shift $A^\nn$, where $A$ is a finite alphabet. Let $X\subset A^\nn$ be a specified sofic subshift. It is well-known that there is a unique Gibbs measure $\mu_\phi$ on $X$…

We establish quantitative results for the statistical be\-ha\-vi\-our of \emph{infinite systems}. We consider two kinds of infinite system: i) a conservative dynamical system $(f,X,\mu)$ preserving a $\sigma$-finite measure $\mu$ such that…

Let $K \subset \mathbb{R}^{2}$ be a rotation and reflection free self-similar set satisfying the strong separation condition, with dimension $\dim K = s > 1$. Intersecting $K$ with translates of a fixed line, one can study the $(s -…

In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form…

In this paper we investigate periodic FPU chains with an even number of particles. We show that near the equilibrium point, any such chain admits a \emph{resonant} Birkhoff normal form of order four which is \emph{completely integrable} -…

We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d \sigma (y)$, where $ \sigma $ is the normalized rotation invariant…

Let $(X_k)_{k\geq 0}$ be a stationary and ergodic process with joint distribution $\mu$ where the random variables $X_k$ take values in a finite set $\mathcal{A}$. Let $R_n$ be the first time this process repeats its first $n$ symbols of…