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

Definition

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.

is called the orbit portrait of the periodic orbit.

must have the same number of elements, which is called the valence of the portrait.

Examples

Parabolic or repelling orbit portrait

valence 2

valence 3

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 :

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 :

valence 4

Formal orbit portraits

has the following properties:

• The doubling map and preserves cyclic order of the angles.

• is called the recurrent ray period.

• where each interval contains one of the sets.

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

, 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

. This is not necessarily distinct from the critical arc.

) are both in every critical arc.

, 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

of the orbit, the external rays. 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.

, the critical value, is in the critical value sector.

Parameter wakes

Two parameter rays land at the same point of the Mandelbrot set in parameter space. A quadratic polynomial for about

Primitive and satellite orbit portraits

is the recurrent ray period, then these two types may be characterized as follows:

• is the base point of a baby Mandelbrot set in parameter space.

• is the base point of a parabolic bifurcation in parameter space.

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.

See also

• Lamination

References