Related papers: Vector valued unified martingale and ergodic theor…

We introduce a new class of sparse sequences that are ergodic and pointwise universally $L^2$-good for ergodic averages. That is, sequences along which the ergodic averages converge almost surely to the projection to invariant functions.…

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…

We prove pointwise and maximal ergodic theorems for probability measure preserving (p.m.p.) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable type $III_1$. We show that this…

We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used the simple fact that the shift map on integers has a simple Lebesgue spectrum. As a…

It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge…

We study optimization problems in ergodic theory from the view point of minimax problems. We give minimax characterizations of maximum ergodic averages involving time averages. Our approach works for the abstract variational principle of…

We obtain ergodic theorems for multiple iterated sums and integrals of the form $\Sigma^{(\nu)}(t)=\sum_{0\leq k_1<...<k_\nu\leq t}\xi(k_1)\otimes\cdots\otimes\xi(k_\nu)$, $t\in[0,T]$ and $\Sigma^{(\nu)}(t)=\int_{0\leq s_1\leq...\leq…

In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$-actions involving averages over dilated submanifolds, thereby generalizing the theory of spherical averages. Our main result is a quantitative estimate for the error term of…

Recently, in \cite{GXHTM}, the authors established $L^p$-boundedness of vector-valued $q$-variational inequalities for averaging operators which take values in the Banach space satisfying martingale cotype $q$ property. In this paper, we…

The main result of this note, Theorem 2, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant under the action of the infinite unitary group and that admits well-defined projections onto the…

For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic theorem with polynomial weights under the coincidence of the quasi discrete spectrums of the system in both senses of Abramov and of Hahn-Parry.…

A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…

We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study…

We introduce a family of pairings between a bounded divergence-measure vector field and a function $u$ of bounded variation, depending on the choice of the pointwise representative of $u$. We prove that these pairings inherit from the…

We study an intermittent quasistatic dynamical system composed of nonuniformly hyperbolic Pomeau--Manneville maps with time-dependent parameters. We prove an ergodic theorem which shows almost sure convergence of time averages in a certain…

We study the joint laws of a continuous, uniformly integrable martingale, its maximum, and its minimum. In particular, we give explicit martingale inequalities which provide upper and lower bounds on the joint exit probabilities of a…

We study the convergence of the so-called entangled ergodic averages $\frac{1}{N^k}\sum_{n_1,...,n_k=1}^{N}T_m^{n_{\alpha(m)}}A_{m-1}T_{m-1}^{n_{\alpha(m-1)}}A_{m-2}...A_1T_1^{n_{\alpha(1)}},$ where $k\leq m$ and…

This is an earlier, but more general, version of "An L^1 Ergodic Theorem for Sparse Random Subsequences". We prove an L^1 ergodic theorem for averages defined by independent random selector variables, in a setting of general…

In this paper we prove a general ergodic theorem for ergodic and measure preserving actions of R^d on standard Borel spaces. In particular, we cover R.L. Jones ergodic theorem on spheres. Our main theorem is concerned with ergodic averages…

We characterize Birkhoff-James orthogonality of continuous vector-valued functions on a compact topological space. As an application of our investigation, Birkhoff-James orthogonality of real bilinear forms are studied. This allows us to…