Related papers: Joint ergodicity of Hardy field sequences
Following an approach presented by N. Frantzikinakis, we prove that any multiple correlation sequence, defined by invertible measure preserving actions of commuting transformations with integer part polynomial iterates, is the sum of a…
Nous nous int\'eressons \`a la convergence vers sa moyenne spatiale ergodique de la moyenne temporelle d'une observable d'un flow hamiltonien \`a un degr\'e et demi de libert\'e avec espace des phases mixte. L'analyse est faite au travers…
In this paper, we study the almost everywhere convergence of sequences of two-parameter ergodic averages over rectangles in the plane. On the one hand, we show that if the rectangles we consider have their sides with slopes in a finitely…
We develop a framework for the study of the limiting behavior of multiple ergodic averages with commuting transformations when all iterates are given by the same sparse sequence; this enables us to partially resolve several longstanding…
The article is devoted to the expansions of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the sense of norm in the space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ The method of generalized multiple…
We show that for every ergodic and aperiodic probability preserving system $(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem. We use the…
We consider a process on $\mathbb{T}^2$, which consists of fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure…
The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin,…
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic. Using methods from ergodic theory, we are able to partially resolve this…
We apply a method inspired by Ratner's work on quantitative mixing for the geodesic flow (Ergod. Theory Dyn. Syst., 1987) and developed by Burger (Duke Math. J., 1990) to study ergodic integrals for horocycle flows. We derive an explicit…
We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As a consequence, we deduce in…
For any $s\in (1/2,1]$, the series$F_s(x)=\sum_{n=1}^{\infty} e^{i\pi n^2 x}/n^s$ converges almost everywhere on $[-1,1]$ by a result of Hardy-Littlewood, but not everywhere. However, there does not yet exist an intrinsic description of the…
In this paper, we reduce pointwise convergence of polynomial ergodic averages of general measure-preserving system acted by $\mathbb{Z}^{d}$ to the case of measure-preserving system acted by $\mathbb{Z}^{d}$ with zero entropy. As an…
A common measure of a function's complexity is the count of its stationary points. For complicated functions, this count grows exponentially with the volume and dimension of their domain. In practice, the count is averaged over a class of…
It is well known that ergodic theory can be used to formally prove a weak form of relaxation to equilibrium for finite, mixing, Hamiltonian systems. In this Letter we extend this proof to any dynamics that preserves a mixing equilibrium…
In this paper, we investigate ergodic and fractal properties of the sets $$\Lambda_y:=\Big\{n\in\mathbb{N}:\ \{u_ny\}\in I_n\Big\},$$ where $\{\cdot\}$ denotes the fractional part function, $(u_n)_{n\in\mathbb{N}}$ is an increasing sequence…
In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation…
In 2022, Bergelson and Richter gave a new dynamical generalization of the prime number theorem by establishing an ergodic theorem along the number of prime factors of integers. They also showed that this generalization holds as well if the…
This text is addressed to students. It is a short story about some problems in ergodic theory, both related and independent. We discuss the factorization of transformations into the product of three involutions; Furstenberg's theorem on…
In this paper we study the multiple ergodic averages $$ \frac{1}{n}\sum_{k=1}^n \varphi(x_k, x_{kq}, ..., x_{k q^{\ell-1}}), \qquad (x_n) \in \Sigma_m $$ on the symbolic space $\Sigma_m ={0, 1, ..., m-1}^{\mathbb{N}^*}$ where $m\ge 2,…