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.Template:Sfn

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

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

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

$x(N_i)\to y.$

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

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

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

• Template:Conway A Course in Functional Analysis
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