A dynamical algorithm to compute hyperbolic Julia sets in polynomial time

TL;DR

This paper introduces a new dynamical algorithm for computing hyperbolic Julia sets of complex polynomials in polynomial time, offering an alternative to previous complex-analytic methods and enabling the computation of the hyperbolicity locus.

Contribution

The paper presents a dynamical approach for polynomial-time computability of hyperbolic Julia sets and establishes lower computability of the hyperbolicity locus, differing from prior complex analysis techniques.

Findings

Algorithm achieves polynomial-time computation of hyperbolic Julia sets.

Establishes lower computability of the hyperbolicity locus.

Provides an alternative dynamical method to classical complex analysis approaches.

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.