Markov–Kakutani fixed-point theorem

In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point.

Statement

Let E be a locally convex topological vector space. Let C be a compact convex subset of E. Let S be a commuting family of self-mappings T of C which are continuous and affine, i.e. T(tx +(1 – t)y) = tT(x) + (1 – t)T(y) for t in [0,1] and x, y in C. Then the mappings have a common fixed point in C.

Proof for a single affine self-mapping

Let T be a continuous affine self-mapping of C.

For x in C define other elements of C by

${\displaystyle x(N)={1 \over N+1}\sum _{n=0}^{N}T^{n}(x).}$

Since C is compact, there is a convergent subnet in C:

${\displaystyle x(N_{i})\rightarrow y.\,}$

To prove that y is a fixed point, it suffices to show that f(Ty) = f(y) for every f in the dual of E. (The dual separates points by the Hahn-Banach theorem; this is where the assumption of local convexity is used.)

Since C is compact, |f| is bounded on C by a positive constant M. On the other hand

${\displaystyle |f(Tx(N))-f(x(N))|={1 \over N+1}|f(T^{N+1}x)-f(x)|\leq {2M \over N+1}.}$

Taking N = N_i and passing to the limit as i goes to infinity, it follows that

${\displaystyle f(Ty)=f(y).\,}$

Hence

${\displaystyle Ty=y.\,}$

Proof of theorem

The set of fixed points of a single affine mapping T is a non-empty compact convex set C^T by the result for a single mapping. The other mappings in the family S commute with T so leave C^T invariant. Applying the result for a single mapping successively, it follows that any finite subset of S has a non-empty fixed point set given as the intersection of the compact convex sets C^T as T ranges over the subset. From the compactness of C it follows that the set

${\displaystyle C^{S}=\{y\in C\mid Ty=y,\,T\in S\}=\bigcap _{T\in S}C^{T}\,}$

is non-empty (and compact and convex).

References

• Markov, A. (1936), "Quelques théorèmes sur les ensembles abéliens", Dokl. Akad. Nauk SSSR, 10: 311–314
• Kakutani, S. (1938), "Two fixed point theorems concerning bicompact convex sets", Proc. Imp. Akad. Tokyo, 14: 242–245
• Reed, M.; Simon, B. (1980), Functional Analysis, Methods of Mathematical Physics, 1 (2nd revised ed.), Academic Press, p. 152, ISBN 0-12-585050-6