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 : Given a quadratic map from the complex plane to itself and a repelling or parabolic periodic orbit
be the set of angles whose corresponding external rays land at
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+0.79111*i portrait of parabolic period 3 orbit is :[1]
Rays for above angles land on points of that orbit .
Parameter c is a center of period 9 hyperbolic component of Mandelbrot set.
of subsets of the circle which satisfy these four properties above is called a formal orbit portrait.
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.
have only a single element are called trivial, except for orbit portrait
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
Trivial orbit portraits are pathological in some respects, and in the sequel we will refer only to nontrivial orbit portraits.
divides the circle into a number of disjoint intervals, called complementary arcs based at the point
The length of each interval is referred to as its angular width.
The critical arc always has length greater than
This is not necessarily distinct from the critical arc.
's, there is a unique smallest 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.
of the orbit, the external rays landing at
open sets called sectors based at
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.
land at the same point of the Mandelbrot set in parameter space if and only if there exists an orbit portrait
be the common landing point of the two external angles in parameter space corresponding to the characteristic arc of
These two parameter rays, along with their common landing point, split the parameter space into two open components.
is realized with a parabolic orbit only for the single value
is the recurrent ray period, then these two types may be characterized as follows: 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.
