GLOBAL ANALYTICAL STRUCTURE OF THE MANDELBROT SET AND ITS GENERALIZATION

In this paper we study the global structure of periodic orbits for a one-dimensional complex map Z(n+1) = Z(n)m + C by the algebraic analytical method advanced by the author in 1985. We give a general formula for the calculation of the orbit number H(N) of any period-N orbit. We also verify rigorously that the complex structures of the Mandelbrot set (m = 2) and its generalized form (m > 2) are composed of infinitely many stable regions of different periodic orbits. We find out that the relation between the stable region number I(N) of the period-N orbit and its orbit number H(N) is exactly I(N) = N . H(N)/m. The algebraic equations of the boundary of each element and the locations of its cusp and center can be given precisely. Furthermore, the cause of the infinitely nested structures for these complex figures are explained.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1992HK81800006&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=8e1609b174ce4e31116a60747a720701Mathematics, AppliedMathematicsSCI(E)0ARTICLE2175-1853