发表机构
Instituto Superior Técnico; Universidade do Algarve（高等技术学院; 阿尔加维大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次研究八维合成代数（para-八元数、Okubo 代数）上的二次动力学，证明 Julia 集对称性由根分解支配，Mandelbrot 集为平面集原像，并揭示双边映射与八元数二次映射的联系。
AI 中文摘要
关于八元数上迭代 $z\mapsto z^2+c$ 的图像已被绘制了三十多年，并且已知八元数 Julia 集是关于常数虚部旋转的旋转体。然而，实数八维合成代数远比八元数丰富：除了 Hurwitz 代数之外，还有对称合成代数、para-八元数和 Okubo 代数，它们没有单位元，其自同构群分别为 $G_2$ 和 $PSU(3)$。在本工作中，我们据我们所知首次研究了这些代数上的二次动力学，并且以实验精神进行：一个包含 134 个四维渲染图、逐季度测量的图集提出了命题，然后我们证明了这些命题并针对新的预测图像进行了测试。我们证明了 Okubo 代数的 Julia 集的对称性由 $\mathfrak{su}(3)$ 关于 $1$ 和 $i$ 平面的根分解所支配：对于该平面中的常数，连续对称性是一个极大环面，它在三个根平面上通过旋转作用，并在 Weyl 壁上增长为 $U(2)$；对称化乘积的自同构群是 $PSU(3)\rtimes\mathbb{Z}_2$。然后我们证明了每个对称合成代数以 $0$ 为种子的 Mandelbrot 集是通过不变量 $n(c)$ 和 $n(c,c\diamond c)$ 对单个平面集的原像：在紧致情形下是 tricorn，在分裂情形下是 tricorn 和双曲 tricorn。这个结果相当令人惊讶，因为 para-八元数和 Okubo 的 Mandelbrot 集被证明是同一个平面图形被两个不同的群饱和。最后，我们证明了每个对称合成代数上的双边映射实际上是八元数二次映射的伪装，并且在 Petersson 角 $\theta=90^\circ$ 时，动力学获得了并非来自自同构的对称性。
英文摘要
The iteration of $z\mapsto z^2+c$ over the octonions has been drawn for more than thirty years, and it is known that the octonionic Julia sets are bodies of revolution about the imaginary part of the constant. Real eight-dimensional composition algebras, however, are much richer than the octonions: beside the Hurwitz algebras there are the symmetric composition algebras, the para-octonions and the Okubo algebras, which have no unit and whose automorphism groups are $G_2$ and $PSU(3)$ respectively. In this work we study, to our knowledge for the first time, the quadratic dynamics over these algebras, and we do it in an experimental spirit: an atlas of 134 four-dimensional renderings, measured quarter by quarter, suggests the statements, which we then prove and test against new, predicted pictures. We show that the symmetries of the Julia sets of the Okubo algebra are governed by the root decomposition of $\mathfrak{su}(3)$ with respect to the plane of $1$ and $i$: for a constant in that plane the continuous symmetry is a maximal torus, which acts by rotations on three root planes and grows to $U(2)$ on the Weyl walls; the automorphism group of the symmetrized product is $PSU(3)\rtimes\mathbb{Z}_2$. We then prove that the Mandelbrot set with seed $0$ of every symmetric composition algebra is the pull-back of a single planar set through the invariants $n(c)$ and $n(c,c\diamond c)$: the tricorn in the compact case, the tricorn and a hyperbolic tricorn in the split case. The result is quite surprising since the para-octonionic and the Okubonic Mandelbrot sets turn out to be the same planar figure saturated by two different groups. Finally, we show that every bilateral map over a symmetric composition algebra is an octonionic quadratic map in disguise, and that at the Petersson angle $θ=90^\circ$ the dynamics acquires symmetries that do not come from automorphisms.
Comments179 pages, 26 figures, 6 tables. The atlas of 134 plates is Appendix B. Scripts and recorded runs are included as ancillary files
