Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A dynamical algorithm to compute hyperbolic Julia sets in polynomial time

Suzanne Boyd, Christian Wolf

TL;DR

This paper presents an alternative method for establishing poly-time computability of hyperbolic Julia sets, which allows to establish, via a new algorithm, lower computability of the hyperbolicity locus of polynomials of a fixed degree.

Abstract

Hyperbolic Julia sets of complex polynomials are known to be computable in polynomial time due to pioneering work of Braverman in 2005 (10.1016/j.entcs.2004.06.031). In this paper, we present an alternative method for establishing poly-time computability of hyperbolic Julia sets, which allows us to establish, via a new algorithm, lower computability of the hyperbolicity locus of polynomials of a fixed degree. We first adapt our recently developed algorithms for the computability of polynomial skew products (preprint available arXiv.2508.08033) and then apply a refinement that allows us to establish poly-time computation of hyperbolic Julia sets. Finally, we derive lower computability of the hyperbolicity locus via an adapted lattice/refinement search algorithm. In contrast to Braverman's 2005 algorithm/proof, our approach is dynamical in nature and does not rely on techniques unique to complex analysis.

A dynamical algorithm to compute hyperbolic Julia sets in polynomial time

TL;DR

This paper presents an alternative method for establishing poly-time computability of hyperbolic Julia sets, which allows to establish, via a new algorithm, lower computability of the hyperbolicity locus of polynomials of a fixed degree.

Abstract

Hyperbolic Julia sets of complex polynomials are known to be computable in polynomial time due to pioneering work of Braverman in 2005 (10.1016/j.entcs.2004.06.031). In this paper, we present an alternative method for establishing poly-time computability of hyperbolic Julia sets, which allows us to establish, via a new algorithm, lower computability of the hyperbolicity locus of polynomials of a fixed degree. We first adapt our recently developed algorithms for the computability of polynomial skew products (preprint available arXiv.2508.08033) and then apply a refinement that allows us to establish poly-time computation of hyperbolic Julia sets. Finally, we derive lower computability of the hyperbolicity locus via an adapted lattice/refinement search algorithm. In contrast to Braverman's 2005 algorithm/proof, our approach is dynamical in nature and does not rely on techniques unique to complex analysis.
Paper Structure (11 sections, 16 theorems, 25 equations, 1 figure, 1 algorithm)

This paper contains 11 sections, 16 theorems, 25 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and background material
  3. Computability
  4. Computability in the complex plane
  5. Poly-time computability.
  6. Polynomial dynamics and invariant sets
  7. Distortion lemmas
  8. Results
  9. Overhead/first step computation
  10. Poly-time computability of $J$
  11. More Proof Details

Key Result

Theorem 1.1

The Julia set of a hyperbolic polynomial of degree $d\geq 2$ is computable to precision $2^N$ in time $O(N\cdot M(N))$.

Figures (1)

  • Figure 1: Left: In Algorithm \ref{['alg:computeJ']}, we consider boxes $\mathcal{U}'$ and their "doubled" boxes with the same center an ideal point $z'$ but double diameter $2\mathcal{U}'$. We also consider Euclidean balls nested between these, so $\mathcal{U}' \subset \Omega_L \subset \Omega_U \subset 2\mathcal{U}'$. Right: Additionally, we consider (hypothetical) smaller balls inside of $\Omega_U$, with center points in $\Omega_L$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 26 more