5 Recurrence 485Furstenberg’s approach to this result is not really shorter than Szemeredi’s, but it is
much more systematic. In fact the following generalization of Furstenberg’s theorem
was given soon afterwards by Furstenberg and Katznelson (1978):
If T 1 ,...,Tpare commuting measure-preserving transformations of the probabil-
ity space(X,B,μ)and if B∈Bwithμ(B)>0,thenμ(B∩T 1 −nB∩···∩T−pnB)> 0
for infinitely many n≥1.
Furstenberg and Katznelson could then deduce quite easily a multi-dimensional
extension of Szemeredi’s theorem which is still beyond the reach of combinatorial
methods. Szemeredi’s theorem was itself afar-reaching generalization of a famous
theorem of van der Waerden (1927):
IfN=S 1 ∪···∪Sris a partition of the set of all positive integers into finitely
many subsets, then one of the subsets Sjcontains arithmetic progressions of arbitrary
finite length.
Szemeredi’s result further indicates how the subsetSjshould be chosen.
Poincar ́e’s measure-theoretic recurrence theorem has a topological counterpart due
to Birkhoff (1912):
If X is a compact metric space and T:X→X a continuous map, then there exists
a point z∈X and an increasing sequence(nk)of positive integers such that Tnkz→z
as k→∞.
Before Furstenberg and Katznelson proved their measure-theoretic theorem,
Furstenberg and Weiss (1978) had already proved its topological counterpart:
If X is a compact metric space and T 1 ,...,Tpcommuting continuous maps of X
into itself, then there exists a point z∈X and an increasing sequence(nk)of positive
integers such that Tinkz→zask→∞(i= 1 ,...,p).
From their theorem Furstenberg and Weiss were able to deduce quite easily both
van der Waerden’s theorem and a known multi-dimensional generalization of it, due
to Gr ̈unwald. It would take too long to prove here Szemeredi’s theorem by the method
of Furstenberg and Katznelson, but we will prove van der Waerden’s theorem by the
method of Furstenberg and Weiss. The proof illustrates how results in one area of
mathematics can find application in another area which is apparently unrelated.
Proposition 26Let ( X ,d) be a compact metric space and T:X→X a continuous
map. Then, for any realε> 0 and any p∈N, there exists some z∈X and n∈N
such that
d(Tnz,z)<ε, d(T^2 nz,z)<ε,...,d(Tpnz,z)<ε.Proof (i) A subsetAofXis said to beinvariantunderTifTA⊆A. The closureA ̄of
an invariant setAis again invariant since, by the continuity ofT,TA ̄⊆TA.LetFbe
the collection of all nonempty closed invariant subsets ofX. ClearlyFis not empty,
sinceX∈F.IfweregardFas partially ordered by inclusion then, byHausdorff ’s
maximality theorem,Fcontains a maximal totally ordered subcollectionT.Thein-
tersectionZof all the subsets inT is both closed and invariant. It is also nonempty,
sinceXis compact. HenceZ∈T and, by construction, no nonempty proper closed
subset ofZis invariant.
By replacingXby its compact subsetZwe may now assume that the only closed
invariant subsets ofXitself areXand∅.