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

Proof for a single affine self-mapping

Let $T$ be a continuous affine self-mapping of $K$.

For $x$ in $K$ define a net $\{x(N)\}_{N\in \mathbb {N} }$ in $K$ by

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

Since $K$ is compact, there is a convergent subnet in $K$:

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

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

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

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

Taking $N=N_{i}$ and passing to the limit as $i$ goes to infinity, it follows that

$f(Ty)=f(y).\,$

Hence

$Ty=y.\,$

Proof of theorem

The set of fixed points of a single affine mapping $T$ is a non-empty compact convex set $K^{T}$ by the result for a single mapping. The other mappings in the family $S$ commute with $T$ so leave $K^{T}$ 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 $K^{T}$ as $T$ ranges over the subset. From the compactness of $K$ it follows that the set

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

is non-empty (and compact and convex).

References

• Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 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, vol. 1 (2nd revised ed.), Academic Press, p. 152, ISBN 0-12-585050-6