Regular branched covering spaces and chaotic maps on the Riemann sphere

Commun. Korean Math. Soc. 2004 Vol. 19, No. 3, 507-517 Printed September 1, 2004

Joo Sung Lee Dongguk University

Abstract : Let $(2,2,2,2)$ be ramification indices for the Riemann sphere. It is well known that the regular branched covering map corresponding to this, is the Weierstrass ${\mathcal P}$ function. Latt\`es \cite{l} gives a rational function $R(z) = {\frac{z^4 +{ \frac1{2}}g_2 z^2 + {\frac1{16}} g^2_2}{4z^3 - g_2z}}$ which is chaotic on $\overline{C}$ and is induced by the Weierstrass ${\mathcal P}$ function and the linear map $L(z)=2z$ on complex plane $C$. It is also known that there exist regular branched covering maps from $T^2$ onto $\bar C$ if and only if the ramification indices are $(2,2,2,2)$, $(2,4,4)$, $(2,3,6)$ and $(3,3,3)$, by the Riemann-Hurwitz formula. In this paper we will construct regular branched covering maps corresponding to the ramification indices $(2,4,4)$, $(2,3,6)$ and $(3,3,$ $3)$, as well as chaotic maps induced by these regular branched covering maps.

Keywords : chaotic map, branched covering space, Weierstrass $P$ function and the Riemann sphere