In his “Iteration of Rational Functions” Beardon defines a critical point of a rational function (mapping the Riemann sphere to itself) as a point such that the function is not injective in any neighborhood of that point. I’m trying to square Beardon’s definition with the other definition, in which a critical point is a point […]

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with the standard chordal metric. The simplest functions I have found posessing these symmetries are rational functions, with zeroes at the vertices of a certain inscribed […]

Let $P$ denote the quotient space obtained by the action of $\mathbb{Z}\backslash2\mathbb{Z}$ by the antipodal map $z\mapsto\frac{1}{z}$ on the riemann sphere $\hat{\mathbb{C}}$ (identified here with $\mathbb{C}\cup\left\{\infty\right\}$). I identify $P$ with the set: $\left\{ z\in\mathbb{C}:\left|z\right|<1\right\} \cup\left\{ e^{it}:0\leq t\leq\pi\right\} $ that is to say, I use elements of the above set as the representatives for the equivalence […]

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