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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps.Complex dynamics the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics specifies more precisely that it is analytic functions whose dynamics it is to study.Any collection {A_1, ldots, A_n} 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.