Markov–Kakutani fixed-point theorem explained

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( λ x + ( 1-λ )y) = λ T(x) + ( 1-λ )T(y) for all λ 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\inN} in K byx(N) = { 1 \over N +1 }\sum ^{N} _{n =0} T ^{n(x).}

Since

K is compact, there is a convergent subnet in K :x(N _{i) →} 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)|\le { 2 M\over N +1 }.

Taking

N = N _{i} and passing to the limit as i goes to infinity, it follows thatf(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 setK ^{S=\{y\in} K\mid Ty = y, T\in S\} = c ap _{T\in} K ^{T}

is non-empty (and compact and convex).