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Decompositions of rational functions over real and complex numbers and a question about invariant curves

Peter Müller

Abstract

We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.

Decompositions of rational functions over real and complex numbers and a question about invariant curves

Abstract

We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.

Paper Structure

This paper contains 6 sections, 14 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment.
  3. Non-circle Jordan curves invariant under a rational function
  4. Proof of Proposition \ref{['P:Gs']}
  5. Proof of Theorems \ref{['T:b']} and \ref{['T:RC']}
  6. Some more examples

Key Result

Theorem 1.2

For every odd prime $\ell$ there are rational functions $f,g\in{\mathbb{C}}(z)$ both of degree $\ell$, such that

Theorems & Definitions (30)

  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 20 more