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 :
Definition.
Given a quadratic map 
from the complex plane to itself 
and a repelling or parabolic periodic orbit formula_3 of formula_4, so that formula_5 (where subscripts are taken 1 + modulo formula_6), let formula_7 be the set of angles whose corresponding external rays land at formula_8.
Then the set formula_9 is called the orbit portrait of the periodic orbit formula_10.
All of the sets formula_7 must have the same number of elements, which is called the valence of the portrait.
Examples.
Parabolic or repelling orbit portrait.
valence 2.
formula_12
formula_13
formula_14
formula_15
valence 3.
Valence is 3 so rays land on each orbit point.
formula_16
For complex quadratic polynomial with c= -0.03111+0.79111*i portrait of parabolic period 3 orbit is :
formula_17
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 + 0.21650635094611*i. 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 :
formula_18
valence 4.
formula_19
Formal orbit portraits.
Every orbit portrait formula_20 has the following properties:
Any collection formula_31 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. Orbit portraits contain dynamical information about how external rays and their landing points map in the plane, but formal orbit portraits are no more than combinatorial objects. Milnor's theorem states that, in truth, there is no distinction between the two.
Trivial orbit portraits.
Orbit portrait where all of the sets formula_7 have only a single element are called trivial, except for orbit portrait formula_33. An alternative definition is that an orbit portrait is nontrivial if it is maximal, which in this case means that there is no orbit portrait that strictly contains it (i.e. there does not exist an orbit portrait formula_34 such that formula_35). It is easy to see that every trivial formal orbit portrait is realized as the orbit portrait of some orbit of the map formula_36, since every external ray of this map lands, and they all land at distinct points of the Julia set. Trivial orbit portraits are pathological in some respects, and in the sequel we will refer only to nontrivial orbit portraits.
Arcs.
In an orbit portrait formula_31, each formula_7 is a finite subset of the circle formula_39, so each formula_7 divides the circle into a number of disjoint intervals, called complementary arcs based at the point formula_8. The length of each interval is referred to as its angular width. 
Each formula_8 has a unique largest arc based at it, which is called its critical arc. The critical arc always has length greater than formula_43
These arcs have the property that every arc based at formula_8, except for the critical arc, maps diffeomorphically to an arc based formula_45, and the critical arc covers every arc based at formula_45 once, except for a single arc, which it covers twice. The arc that it covers twice is called the critical value arc for formula_45. This is not necessarily distinct from the critical arc.
When formula_48 escapes to infinity under iteration of formula_49, or when formula_48 is in the Julia set, then formula_48 has a well-defined external angle. Call this angle formula_52. formula_52 is in every critical value arc. Also, the two inverse images of formula_48 under the doubling map (formula_55 and formula_56) are both in every critical arc.
Among all of the critical value arcs for all of the formula_7's, there is a unique smallest critical value arc formula_58, called the characteristic arc which is strictly contained within every other critical value arc. The characteristic arc is a complete invariant of an orbit portrait, in the sense that two orbit portraits are identical if and only if they have the same characteristic arc.
Sectors.
Much as the rays landing on the orbit divide up the circle, they divide up the complex plane. For every point formula_8 of the orbit, the external rays landing at formula_8 divide the plane into formula_61 open sets called sectors based at formula_8. Sectors are naturally identified the complementary arcs based at the same point. The angular width of a sector is defined as the length of its corresponding complementary arc. Sectors are called critical sectors or critical value sectors when the corresponding arcs are, respectively, critical arcs and critical value arcs.
Sectors also have the interesting property that formula_63 is in the critical sector of every point, and formula_48, the critical value of formula_49, is in the critical value sector.
Parameter wakes.
Two parameter rays with angles formula_66 and formula_67 land at the same point of the Mandelbrot set in parameter space if and only if there exists an orbit portrait formula_68 with the interval formula_69 as its characteristic arc. For any orbit portrait formula_68 let formula_71 be the common landing point of the two external angles in parameter space corresponding to the characteristic arc of formula_68. These two parameter rays, along with their common landing point, split the parameter space into two open components. Let the component that does not contain the point formula_63 be called the formula_68-wake and denoted as formula_75. A quadratic polynomial formula_76 realizes the orbit portrait formula_20 with a repelling orbit exactly when formula_78. formula_20 is realized with a parabolic orbit only for the single value formula_80
for about
Primitive and satellite orbit portraits.
Other than the zero portrait, there are two types of orbit portraits: primitive and satellite. If 
formula_61 is the valence of an orbit portrait formula_68 and formula_28 is the recurrent ray period, then these two types may be characterized as follows:
Generalizations.
Orbit portraits turn out to be useful combinatorial objects in studying the connection between the dynamics and the parameter spaces of other families of maps as well. In particular, they have been used to study the patterns of all periodic dynamical rays landing on a periodic cycle of a unicritical anti-holomorphic polynomial.