Self-similarity and Julia Sets

Rongwei Yang
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Rongwei Yang: University at Albany, the State University of New York, Department of Mathematics & Statistics

Chapter Chapter 10 in A Spectral Theory Of Noncommuting Operators, 2024, pp 235-257 from Springer

Abstract: Abstract The symmetry of a measure space ( X , μ ) $$(X, \mu )$$ is described by measure-preserving group actions it admits. Such group actions naturally give rise to a unitary representation, called the Koopman representation, of the group on L 2 ( X , μ ) $$L^2(X, \mu )$$ . On the other hand, the symmetry of ( X , μ ) $$(X, \mu )$$ also enables the construction of groups with certain desired properties. Of particular interest is the case when X = [ 0 , 1 ] $$X=[0, 1]$$ with Lebesgue measure μ $$\mu $$ , through which the first example of a group of intermediate growth, the Grigorchuk group G $$\mathcal {G}$$ , was constructed [103]. Subsequently, it was discovered that G $$\mathcal {G}$$ displays a self-similarity property.

Date: 2024
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DOI: 10.1007/978-3-031-51605-4_10

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