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arXiv · 2510.22929

A dynamical algorithm to compute hyperbolic Julia sets in polynomial time

Suzanne Boyd·Christian Wolf
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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.

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Suzanne Boyd, Christian Wolf. 2025-10-27. A dynamical algorithm to compute hyperbolic Julia sets in polynomial time. https://arxiv.org/abs/2510.22929

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