Advertisement

The Markov–Kakutani Theorem

Fixed Points For Commuting Families Of Affine Maps
  • Joel H. Shapiro
Chapter
Part of the Universitext book series (UTX)

Overview

Consider the unit circle, the set \(\mathbb{T}\) of complex numbers of modulus one. Complex multiplication makes \(\mathbb{T}\) into a group, and the topology inherited from the complex plane makes it into a compact metric space. Here topology and algebra complement each other in that the group operations of multiplication \(\mathbb{T} \times \mathbb{T} \rightarrow \mathbb{T}\) and inversion \(\mathbb{T} \rightarrow \mathbb{T}\) are continuous. Tied up with the topology and algebra of \(\mathbb{T}\) is arc-length measure defined on the Borel subsets of \(\mathbb{T}\), the salient property of which is its rotation invariance: \(\sigma (\gamma E) =\sigma (E)\) for each \(\gamma \in \mathbb{T}\) and Borel subset E of \(\mathbb{T}\).

Keywords

Compact Group Haar Measure Topological Vector Space Common Fixed Point Borel Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 15.
    Boyce, W.M.: Commuting functions with no common fixed point. Trans. Amer. Math. Soc. 137, 77–92 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 25.
    Chernoff, P.R.: A simple proof of Tychonoff’s theorem via nets. Amer. Math. Monthly 99, 932–934 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 52.
    Huneke, J.P.: On common fixed points of commuting continuous functions on an interval. Proc. Amer. Math. Soc. 139, 371–381 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 55.
    Jech, T.J.: The Axiom of Choice. North Holland, Amsterdam (1973). Republished by Dover, New York (2008)Google Scholar
  5. 56.
    Kakutani, S.: Two fixed-point theorems concerning bicompact convex sets. Proc. Imp. Acad. (Jap.) 14, 242–245 (1938)Google Scholar
  6. 75.
    Markov, A.A.: Quelques théoremes sur les ensembles abéliens. C.R. URSS 2, p. 311 (1936)Google Scholar
  7. 97.
    Riesz, F.: Sur les opérations fonctionelles linéaires. C. R. Math. Acad. Sci. Paris 149, 974–977 (1909)zbMATHGoogle Scholar
  8. 100.
    Rudin, W.: Fourier Analysis on Groups. Interscience, vol. 12 Wiley, New York (1962)Google Scholar
  9. 101.
    Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)zbMATHGoogle Scholar
  10. 103.
    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  11. 120.
    Tychonoff, A.: Ein Fixpunktsatz. Math. Ann. 111, 767–776 (1935)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Joel H. Shapiro
    • 1
  1. 1.Portland State UniversityPortlandUSA

Personalised recommendations