Nonconventional ergodic averages and nilmanifolds

  title={Nonconventional ergodic averages and nilmanifolds},
  author={Bernard Host and Bryna Kra},
  journal={Annals of Mathematics},
We study the L2-convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg?fs proof of Szemer?Ledi?fs theorem. The second average is taken along cubes whose sizes tend to +??. For each average, we show that it is sufficient to prove the convergence for special systems, the characteristic factors. We build these factors in a general way, independent of… 

Pointwise convergence in nilmanifolds along smooth functions of polynomial growth

We study the equidistribution of orbits of the form b a1(n) 1 · · · b ak(n) k Γ in a nilmanifold X, where the sequences ai(n) arise from smooth functions of polynomial growth belonging to a Hardy

Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations

We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iterates

Multiple ergodic averages for flows and an application

We show the $L^2$-convergence of continuous time ergodic averages of a product of functions evaluated at return times along polynomials. These averages are the continuous time version of the averages

Pointwise convergence for cubic and polynomial multiple ergodic averages of non-commuting transformations

Abstract We study the limiting behavior of multiple ergodic averages involving several, not necessarily commuting, measure-preserving transformations. We work on two types of averages, one that uses

Complexity of nilsystems and systems lacking nilfactors

Nilsystems are a natural generalization of rotations and arise in various contexts, including in the study of multiple ergodic averages in ergodic theory, in the structural analysis of topological

Convergence of diagonal ergodic averages

  • H. Towsner
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2009
Abstract Tao has recently proved that if T1,…,Tl are commuting, invertible, measure-preserving transformations on a dynamical system, then for any L∞ functions f1,…,fl, the average (1/N)∑ n=0N−1∏

Ergodic averages of commuting transformations with distinct degree polynomial iterates

We prove mean convergence, as N → ∞, for the multiple ergodic averages 1N∑n=1Nf1(T1p1(n)x)⋅…⋅fl(Tlplx) where p1, …, pℓ are integer polynomials with distinct degrees, and T1, …, Tℓ are commuting,

Ergodicity of the Liouville system implies the Chowla conjecture

The Chowla conjecture asserts that the values of the Liouville function form a normal sequence of plus and minus ones. Reinterpreted in the language of ergodic theory it asserts that the Liouville

A multidimensional Szemerédi theorem for Hardy sequences of different growth

Abstract. We prove a variant of the multidimensional polynomial Szemeredi theorem of Bergelson and Leibman where one replaces polynomial sequences with other sparse sequences defined by functions



An odd Furstenberg-Szemerédi theorem and quasi-affine systems

We prove a version of Furstenberg’s ergodic theorem with restrictions on return times. More specifically, for a measure preserving system (X, B, μ,T), integers 0 ≤j 0, we show that there existsn ≡ j

Strict Ergodicity and Transformation of the Torus

Introduction. If T is a measure preserving transformation ofl a probability space Q with measure Iu, the ergodic theorem assures the existence N-1 almost everywhere with respect to /i of the average

Pointwise ergodic theorems for arithmetic sets

converge almost surely for N -+ co, assuming f a function of class L~(~, ~). Here and in the sequel, one denotcs by ~ a probability measure and by T a measure-preserving automorphism. The natural

Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold

  • A. Leibman
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2004
We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This

Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques

  • E. Lesigne
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1991
Abstract We call nilmanifold every compact space X on which a connected locally compact nilpotent group acts transitively. We show that, if X is a nilmanifold and f is a continuous function on X,

On Raghunathan’s measure conjecture

This paper represents the last in our three-part series on Raghunathan's measure conjecture (see [R1], [R2] for Parts I and II). More specifically, let G be a real Lie group (all groups in this paper

A new proof of Szemerédi's theorem

In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these

Topological Transformation Groups

1. Introduction This note will summarize some of the recent work on topological groups and discuss a few topics in transformation groups mainly in S 3 and S 4. In one aspect of this subject, namely

Sur un théorème ergodique pour des mesures diagonales

En reprenant l'etude d'une equation fonctionnelle associee a la convergence des moyennes ergodiques de la forme 1/nΣ n=0 N−1 f(T n x)•g(T 2n x)•h(T 3n x) (1) on montre que le comportement de (1) fait

Eine Verschärfung des Poincaréschen “Wiederkehrsatzes”

© Foundation Compositio Mathematica, 1935, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions