A dynamical algorithm to compute hyperbolic Julia sets in polynomial time
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910028551684096 |
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| author | Boyd, Suzanne Wolf, Christian |
| author_facet | Boyd, Suzanne Wolf, Christian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A dynamical algorithm to compute hyperbolic Julia sets in polynomial time Boyd, Suzanne Wolf, Christian Dynamical Systems Primary: 37F10, 03D15, Secondary: 03D80, 37F15 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. |
| title | A dynamical algorithm to compute hyperbolic Julia sets in polynomial time |
| topic | Dynamical Systems Primary: 37F10, 03D15, Secondary: 03D80, 37F15 |
| url | https://arxiv.org/abs/2510.22929 |