Preperiodicity and systematic extraction of periodic orbits of the quadratic map

TL;DR

This paper introduces a method to systematically extract exact equations of periodic orbits of the quadratic map using preperiodic points, avoiding complex polynomial explosion and revealing structural insights into algebraic number towers.

Contribution

It presents a novel approach leveraging preperiodic points to derive exact orbital equations without iterative polynomial explosion.

Findings

Exact equations for periodic orbits can be obtained systematically.

The method reveals the arithmetic structure of algebraic number towers.

Provides insights into bifurcation cascades and nesting properties.

Abstract

Iteration of the quadratic map produces sequences of polynomials whose degrees {\sl explode} as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree $4020$, while for the $52,377$ period-20 orbits the degree rises already to $1,047,540$. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, with no 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.