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. This theorem is a key tool in one of the quickest proofs of amenability of abelian groups.

Statement

Let [math]\displaystyle{ X }[/math] be a locally convex topological vector space, with a compact convex subset [math]\displaystyle{ K }[/math]. Let [math]\displaystyle{ S }[/math] be a family of continuous mappings of [math]\displaystyle{ K }[/math] to itself which commute and are affine, meaning that [math]\displaystyle{ T(\lambda x + (1-\lambda)y) = \lambda T(x) + (1-\lambda)T(y) }[/math] for all [math]\displaystyle{ \lambda }[/math] in [math]\displaystyle{ (0,1) }[/math] and [math]\displaystyle{ T }[/math] in [math]\displaystyle{ S }[/math]. Then the mappings in [math]\displaystyle{ S }[/math] share a fixed point.

Proof for a single affine self-mapping

Let [math]\displaystyle{ T }[/math] be a continuous affine self-mapping of [math]\displaystyle{ K }[/math].

For [math]\displaystyle{ x }[/math] in [math]\displaystyle{ K }[/math] define a net [math]\displaystyle{ \{x(N)\}_{N\in\mathbb{N}} }[/math] in [math]\displaystyle{ K }[/math] by

[math]\displaystyle{ x(N)={1\over N+1}\sum_{n=0}^N T^n(x). }[/math]

Since [math]\displaystyle{ K }[/math] is compact, there is a convergent subnet in [math]\displaystyle{ K }[/math]:

[math]\displaystyle{ x(N_i)\rightarrow y. \, }[/math]

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

Since [math]\displaystyle{ K }[/math] is compact, [math]\displaystyle{ |f| }[/math] is bounded on [math]\displaystyle{ K }[/math] by a positive constant [math]\displaystyle{ M }[/math]. On the other hand

[math]\displaystyle{ |f(Tx(N))-f(x(N))|={1\over N+1} |f(T^{N+1}x)-f(x)|\le {2M\over N+1}. }[/math]

Taking [math]\displaystyle{ N = N_i }[/math] and passing to the limit as [math]\displaystyle{ i }[/math] goes to infinity, it follows that

[math]\displaystyle{ f(Ty) = f(y). \, }[/math]

Hence

[math]\displaystyle{ Ty = y. \, }[/math]

Proof of theorem

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

[math]\displaystyle{ K^S=\{y\in K\mid Ty=y, \, T\in S\}=\bigcap_{T\in S} K^T \, }[/math]

is non-empty (and compact and convex).

References

• Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. 
• 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