# Nonconventional ergodic averages and nilmanifolds

@article{Host2005NonconventionalEA, title={Nonconventional ergodic averages and nilmanifolds}, author={Bernard Host and Bryna Kra}, journal={Annals of Mathematics}, year={2005}, volume={161}, pages={397-488} }

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…

## 391 Citations

### Pointwise convergence in nilmanifolds along smooth functions of polynomial growth

- Mathematics
- 2022

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

- Mathematics
- 2010

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

- Mathematics
- 2009

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

- MathematicsErgodic Theory and Dynamical Systems
- 2011

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

- Mathematics
- 2012

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

- MathematicsErgodic 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

- Mathematics
- 2011

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

- Mathematics
- 2016

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

- Mathematics
- 2012

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…

## References

SHOWING 1-10 OF 37 REFERENCES

### An odd Furstenberg-Szemerédi theorem and quasi-affine systems

- Mathematics
- 2002

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

- Mathematics
- 1961

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

- Mathematics
- 1989

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

- MathematicsErgodic 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

- MathematicsErgodic 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

- Mathematics
- 1991

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

- Mathematics
- 2001

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

- Mathematics
- 1956

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

- Mathematics
- 1988

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”

- Mathematics
- 1935

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