# 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.

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