In mathematics, the Markov–Kakutani fixed-point theorem, named after

Andrey Markov Andrey Andreyevich Markov, first name also spelled "Andrei", in older works also spelled Markoff) (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research lat ...

and

Shizuo Kakutani was a Japanese-American mathematician, best known for his eponymous fixed-point theorem. Biography Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. At one point he spent two years at the Institute for ...

, states that a commuting family of continuous affine self-mappings of a compact convex subset in a

locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topologica ...

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

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 ,1and ''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 :

x(N)=\sum_^N T^n(x).

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

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 :

, f(Tx(N))-f(x(N)), = , f(T^x)-f(x), \le .

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 ''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 In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...

of ''C'' it follows that the set :

C^S=\=\bigcap_ C^T \,

is non-empty (and compact and convex).

References

* * * Fixed-point theorems {{Improve categories, date=February 2022