Ornstein-Weiss lemma

Let G be a group. For a fixed KG, define the K-boundary of UG as

KU={gGKgU,Kg(GU)}. (1)

Let 𝒫(G) be the set of finite subsets of G. Call a Følner net for G a net 𝒳={Xi}i𝒫(G), being a directed set, such that for every finite KG,

limi|KXi||Xi|=0, (2)

where the limit is taken in the sense of directed sets. Recall that G has a Følner net if and only if G is amenable.

Theorem 1 (Ornstein-Weiss lemma)

Let G be an amenable group and F:PF(G)R a subadditive, right-invariant function, that is:

  1. 1.

    For any two finite subsets U,V of G,

    F(UV)F(U)+F(V). (3)
  2. 2.

    For any gG and finite UG,

    F(Ug)=F(U). (4)

Then for any Følner net X={Xi}iI on G, the limit

L=limiF(Xi)|Xi| (5)

exists, and does not depend on the choice of X.

The Ornstein-Weiss lemma allows to prove variants of Birkhoff’s ergodic theorem for actions of amenable groups, rather than only those generated by an invertiblePlanetmathPlanetmath, measure invariant map. Moreover, it shares several similarities with Fekete’s lemma on subadditive functions over the positive integers, although it is not a completePlanetmathPlanetmath counterpart. In fact, putting Xn={1,,n} determines a Følner sequence on ; however, if f:[0,) is subadditive, then F(U)=f(|U|) is right-invariant, but not necessarily subadditive. (Counterexample: f(n)=nmod 2, U={1,2}, V={2,3}.)


  • 1 Gromov, M. (1999) Topological invariants of dynamical systemsMathworldPlanetmathPlanetmath and spaces of holomorphic maps, I. Math. Phys. Anal. Geom. 2, 323–415.
  • 2 Krieger, F. (2007) Le lemme d’Ornstein-Weiss d’après Gromov. In B. Hasselblatt (ed.), Dynamics, Ergodic Theory, and Geometry. Cambridge University Press.
  • 3 Krieger, F. The Ornstein-Weiss lemma for discrete amenable groups. Preprint.
  • 4 Ornstein, D.S. and Weiss, B. (1987) EntropyPlanetmathPlanetmath and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48, 1–141.
Title Ornstein-Weiss lemma
Canonical name OrnsteinWeissLemma
Date of creation 2013-03-22 19:20:24
Last modified on 2013-03-22 19:20:24
Owner Ziosilvio (18733)
Last modified by Ziosilvio (18733)
Numerical id 5
Author Ziosilvio (18733)
Entry type Theorem
Classification msc 43A07