Related papers: Notes on Austin's multiple ergodic theorem

We establish a "diagonal" ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg's correspondence principle, prove that any "large" set in $K$ contains…

Fixed point theorems are ubiquitous in economic research. Many studies cite Smithson (1971) ``Fixed points of order preserving multifunctions,'' yet the original proof contains errors. This note presents a new, concise proof and explains…

We present a lecture note on Thouvenot's proof of the Roth-Furstenberg theorem and joining proofs of Furstenberg's theorems on multiple progression average mixing for weakly mixing transformations.

Our goal in the present paper is to give a new ergodic proof of a well-known Veech's result, build upon our previous works.

This brief pedagogical note re-proves a simple theorem on the convergence, in $L_2$ and in probability, of time averages of non-stationary time series to the mean of expectation values. The basic condition is that the sum of covariances…

Here I prove some extension theorem for multifunctions in a space with an arbitrary uniform structure and orbital completeness. The motivation comes from a fixed point theorem due to Dhage which is proved as a special case of the theorem…

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good…

We study almost sure limiting behavior of extreme and intermediate order statistics arising from strictly stationary sequences. First, we provide sufficient dependence conditions under which these order statistics converges almost surely to…

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…

We give an account of our work on transitive factorizations of permutations. The work has had impact upon other areas of mathematics such as the enumeration of graph embeddings, random matrices, branched covers, and the moduli spaces of…

I extend further, using new proofs, two generalizations of an earlier orbits-fixed-points theorem, which was restricted to group action of the symmetric group. The extended equality makes use of the Stirling numbers of the second kind. An…

Extended Thermodynamics is a very important theory: for example, it predicts hyperbolicity, finite speeds of propagation waves as well as continuous dependence on initial data. Therefore, it constitutes a significative improvement of…

Mixing for measure-preserving group actions is a fundamental notion in ergodic theory, with different phenomena arising for different acting groups. In 1993, Klaus Schmidt and Tom Ward proved that 2-mixing implies mixing of all orders for…

The semi-invertible version of Oseledets' multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a…

In this paper, we prove that ergodic point processes with moments of all orders, driven by particular infinite measure preserving transformations, have to be a superposition of shifted Poisson processes. This rigidity result has a lot of…

We show that for every ergodic system $(X,\mu,T_1,\ldots,T_d)$ with commuting transformations, the average \[\frac{1}{N^{d+1}} \sum_{0\leq n_1,\ldots,n_d \leq N-1} \sum_{0\leq n\leq N-1} f_1(T_1^n \prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n…

We prove an $r$-variation estimate, $r>4$, in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all $r\ge 2$ as in the analogous cases for one or two commuting…

In this note we present a proof of multiple recurrence for ergodic systems (and thereby of Szemer\'edi's theorem) being a mixture of three known proofs. It is based on a conditional version of the Jacobs-de Leeuw-Glicksberg decomposition…

We prove a version of pointwise Ergodic Theorem for non-stationary random dynamical systems. Also, we discuss two specific examples where the result is applicable: non-stationary iterated function systems and non-stationary random matrix…

This is an expository article/encyclopedia entry explaining the history, techniques, and central results in the field of smooth ergodic theory.