Fixed Points for Non-commuting Map Families
Here we’ll generalize the Markov–Kakutani Theorem (Theorem 9.6, p. 107) to collections of affine, continuous maps that obey a generalized notion of commutativity inspired by the group-theoretic concept of solvability. This will enable us to show, for example, that the unit disc is not paradoxical even with respect to its full isometry group, and that solvable groups are amenable, hence not paradoxical. We’ll prove that compact solvable groups possess Haar measure, and will show how to extend this result to solvable groups that are just locally compact.
KeywordsCompact Group Composition Operator Haar Measure Radon Measure Solvable Group
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