Preperiodicity and systematic extraction of periodic orbits of the quadratic map

Abstract

Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree [Formula: see text], while for the [Formula: see text] period-20 orbits the degree rises already to [Formula: see text]. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map.