The Markov–Kakutani Theorem

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


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 \(\mathbb{T}\).


Compact Group Haar Measure Topological Vector Space Common Fixed Point Borel Subset 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

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