REGULAR BRANCHED COVERING SPACES AND CHAOTIC MAPS ON THE RIEMANN SPHERE

Title & Authors
REGULAR BRANCHED COVERING SPACES AND CHAOTIC MAPS ON THE RIEMANN SPHERE
Lee, Joo-Sung;

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 P function. Lattes [7] gives a rational function R(z)= $\small{{\frac{z^4+{\frac{1}{2}}g2^{z}^2+{\frac{1}{16}}g{\frac{2}{2}}}$ which is chaotic on $\small{{\bar{C}}}$ 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 $\small{T^2}$ onto $\small{{\bar{C}}}$ 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;the Riemann sphere;
Language
English
Cited by
1.
A CHARACTERIZATION OF HYPERBOLIC TORAL AUTOMORPHISMS,;

대한수학회논문집, 2006. vol.21. 4, pp.759-769
References
1.
Math. Tidsskrift, 1951. vol.B. pp.56-58

2.
An Introduction to Chaotic Dynamical Systems, 1987.

3.
Acta Math., 1993. vol.171. pp.263-297

4.
London Math. Soc. Lecture Notes Series 9, 1973.

5.
Math. Tidsskrift, 1952. vol.B. pp.61-65

6.
The Functions of Mathematical Physics, 1971.

7.
CR Acad. Sci. Paris, 1918. vol.166. pp.26-28

8.
Dynamics in One Complex Variable, 1999.

9.
Pitman Research Notes in Math. Series 161, 1987.

10.
Transl. Math. Monogr., 1997.