Analytic Composition Expansions About Functional Equation Fixed Points

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Analytic Composition Expansions About Functional Equation Fixed Points

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Vogt, Cameron

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2021

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Given holomorphic functions satisfying the functional equation φ = σ ◦ φ ◦ τ where τ has an attracting fixed point paired with a repelling fixed point of σ, we prove φ can be expressed as a composition expansion limn→∞ σ ◦ n ◦ ψ ◦ τ ◦ n where ψ approximates φ in some sense. With certain restrictions, φ is the unique function satisfying the functional equation. Conversely, given a functional equation of the specified form, we construct a function which satisfies it. The idea behind the proof is to view the transformation f → σ ◦ f ◦ τ as a contraction mapping on a particular space of holomorphic functions. As a basic example, the functional equation cos z = 2 cos2 (z/2) − 1 generates a composition expansion for cos z.

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Summa Cum Laude

College of Science and Engineering

Mathematics

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Vogt, Cameron. (2021). Analytic Composition Expansions About Functional Equation Fixed Points. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/220248.

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