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Nonconventional ergodic averages and nilmanifolds

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… Expand

Multiple recurrence and nilsequences

- V. Bergelson, B. Host, Bryna Kra, I. Ruzsa
- Mathematics
- 1 March 2005

Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving system $(X,\mathcal{X},\mu,T)$ and a set… Expand

Substitutional dynamical systems, Bratteli diagrams and dimension groups

- F. Durand, B. Host, Christian F. Skau
- Mathematics, Physics
- 1 August 1999

The present paper explores substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit,… Expand

Ergodic seminorms for commuting transformations and applications

- B. Host
- Mathematics
- 22 November 2008

Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although… Expand

Nilsequences and a structure theorem for topological dynamical systems

Abstract We characterize inverse limits of nilsystems in topological dynamics, via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure… Expand

Uniformity seminorms on ℓ∞ and applications

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by… Expand

Higher order Fourier analysis of multiplicative functions and applications

- N. Frantzikinakis, B. Host
- Mathematics
- 4 March 2014

We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic and… Expand

Asymptotic pairs in positive-entropy systems

- F. Blanchard, B. Host, Sylvie Ruette
- Mathematics
- Ergodic Theory and Dynamical Systems
- 1 June 2002

We show that in a topological dynamical system (X,T) of positive entropy there exist proper (positively) asymptotic pairs, that is, pairs (x,y) such that x\not= y and \lim_{n\to +\infty} d(T^n x,T^n… Expand

Convergence of polynomial ergodic averages

We prove theL2 convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a… Expand

Ergodic averages of commuting transformations with distinct degree polynomial iterates

- Q. Chu, N. Frantzikinakis, B. Host
- Mathematics
- 14 December 2009

We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)\cdot\ldots\cdot f_\ell(T_\ell^{p_\ell(n)}x)$, where $p_1,\ldots,p_\ell$ are… Expand

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