Orbit Portraits in non-Autonomous Iteration

TL;DR

This paper generalizes orbit portraits to non-autonomous polynomial iteration, revealing new phenomena such as irrational-length arcs when polynomial degrees vary, extending classical dynamical systems theory.

Contribution

It introduces a new framework for orbit portraits in non-autonomous iteration and characterizes their properties for constant and varying polynomial degrees.

Findings

Portraits for constant degree sequences are eventually periodic.

Varying degrees can produce portraits with irrational-length complementary arcs.

The work bridges classical and non-autonomous polynomial dynamics.

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.