Orbit portraits in non-autonomous iteration

Abstract

We extend the definition of an orbit portrait to the context of non-autonomous iteration, both for the combinatorial version involving collections of angles, and for the dynamic version involving external rays where combinatorial portraits can be realized by the dynamics associated with sequences of polynomials with suitably uniformly bounded degrees and coefficients. We show that, in the case of sequences of polynomials of constant degree, the portraits which arise are eventually periodic which is somewhat similar to the classical theory of polynomial iteration. However, if the degrees of the polynomials in the sequence are allowed to vary, one can obtain portraits with complementary arcs of irrational length which are fundamentally different from the classical ones.