Orbit portrait

In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps. In simple words one can say that it is : a list of external angles for which rays land on points of that orbit graph showing above list Given a quadratic map from the complex plane to itself and a repelling or parabolic periodic orbit of , so that (where subscripts are taken 1 + modulo ), let be the set of angles whose corresponding external rays land at . Then the set is called the orbit portrait of the periodic orbit . All of the sets must have the same number of elements, which is called the valence of the portrait. Valence is 3 so rays land on each orbit point. For complex quadratic polynomial with c= -0.03111+0.79111i portrait of parabolic period 3 orbit is : Rays for above angles land on points of that orbit . Parameter c is a center of period 9 hyperbolic component of Mandelbrot set. For parabolic julia set c = -1.125 + 0.21650635094611i. It is a root point between period 2 and period 6 components of Mandelbrot set. Orbit portrait of period 2 orbit with valence 3 is : Every orbit portrait has the following properties: Each is a finite subset of The doubling map on the circle gives a bijection from to and preserves cyclic order of the angles. All of the angles in all of the sets are periodic under the doubling map of the circle, and all of the angles have the same exact period. This period must be a multiple of , so the period is of the form , where is called the recurrent ray period. The sets are pairwise unlinked, which is to say that given any pair of them, there are two disjoint intervals of where each interval contains one of the sets. Any collection of subsets of the circle which satisfy these four properties above is called a formal orbit portrait. It is a theorem of John Milnor that every formal orbit portrait is realized by the actual orbit portrait of a periodic orbit of some quadratic one-complex-dimensional map.