# Ornstein-Weiss lemma

Let $G$ be a group. For a fixed $K\subseteq G$, define the $K$-boundary of $U\subseteq G$ as

 $\partial_{K}U=\left\{g\in G\mid Kg\cap U,Kg\cap(G\setminus U)\neq\emptyset% \right\}\;.$ (1)

Let $\mathcal{PF}(G)$ be the set of finite subsets of $G$. Call a Følner net for $G$ a net $\mathcal{X}=\{X_{i}\}_{i\in\mathcal{I}}\subseteq\mathcal{PF}(G),$ $\mathcal{I}$ being a directed set, such that for every finite $K\subseteq G$,

 $\lim_{i\in\mathcal{I}}\frac{|\partial_{K}X_{i}|}{|X_{i}|}=0\;,$ (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 $F:\mathcal{PF}(G)\to\mathbb{R}$ a subadditive, right-invariant function, that is:

1. 1.

For any two finite subsets $U,V$ of $G$,

 $F(U\cup V)\leq F(U)+F(V)\;.$ (3)
2. 2.

For any $g\in G$ and finite $U\subseteq G$,

 $F(Ug)=F(U)\;.$ (4)

Then for any Følner net $\mathcal{X}=\left\{X_{i}\right\}_{i\in\mathcal{I}}$ on $G$, the limit

 $L=\lim_{i\in\mathcal{I}}\frac{F(X_{i})}{|X_{i}|}$ (5)

exists, and does not depend on the choice of $\mathcal{X}$.

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 invertible, measure invariant map. Moreover, it shares several similarities with Fekete’s lemma on subadditive functions over the positive integers, although it is not a complete counterpart. In fact, putting $X_{n}=\{1,\ldots,n\}$ determines a Følner sequence on $\mathbb{Z}$; however, if $f:\mathbb{N}\to[0,\infty)$ is subadditive, then $F(U)=f(|U|)$ is right-invariant, but not necessarily subadditive. (Counterexample: $f(n)=n\,\mathrm{mod}\,2$, $U=\{1,2\}$, $V=\{2,3\}$.)

## References

• 1 Gromov, M. (1999) Topological invariants of dynamical systems 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) Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48, 1–141.
Title Ornstein-Weiss lemma OrnsteinWeissLemma 2013-03-22 19:20:24 2013-03-22 19:20:24 Ziosilvio (18733) Ziosilvio (18733) 5 Ziosilvio (18733) Theorem msc 43A07