Regular branched covering spaces and chaotic maps on the Riemann sphere

Abstract

Let (2; 2; 2; 2) be rami¯cation indices for the Riemann sphere. It is well known that the regular branched covering map corresponding to this, is the Weierstrass P function. Lattµes [7] gives a rational function R(z) = z4+1 2 g2z2+ 1 16 g2 2 4z3¡g2z which is chaotic on C and is induced by the Weierstrass 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 T2 onto ¹ C if and only if the rami¯cation 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 rami¯cation indices (2; 4; 4), (2; 3; 6) and (3; 3; 3), as well as chaotic maps induced by these regular branched cov- ering maps.