Centers of hyperbolic components in the Mandelbrot set, up to period 32, rounded to nearest 10^{-26}

The Mandelbrot set is the set of parameters c such that the sequence z_{n+1} = z_n^2 + c starting from z_0=0 remains bounded. Centers of hyperbolic components are the parameters c for which the sequence (z_n) is periodic. These centers are classified by their period, which is the smallest positive integer k such that z_k=0; the set of such parameters is denoted by Hyp(k). For a given integer n, the roots of the polynomial p_n(c) defined recursively by p_n(c) = p_{n-1}^2(c) + c where p_0(c)=0 is composed of hyperbolic centers of period k, for all divisors k of the integer n. In other terms, the set of roots of p_n is the union of Hyp(k) when k divides n. This data set contains a list of all hyperbolic centers Hyp(k) in the Mandelbrot set, up to period k=32, with non-negative imaginary part. Each file contains a list of roots, z=a+ib with a and b rounded to the nearest 10^{-26}, that is 85 bits of precision. The files are in CSV format, with one root per row, presented as the character chain "a, b". Note b is always non-negative. For periods 26 to 32, the CSV files are compressed using the BZ2 algorithm. This data results from a conversion to human-readable decimal form of a high-precision and certified database, as explained in the reference [arXiv:2402.06083]. This larger database extends all the way to period 41. Please contact the authors if you need access. For pre-periodic parameters, please refer to the dataset [https://doi.org/10.5281/zenodo.15527762]. Note: please be careful that some programs mishandle CSV files that contain floating point numbers whose precision exceed machine precision (which is definitly the case here). To avoid such unpleasantness, we suggest accessing the CSV files programmatically, as text-based files. To add data protection to your workflow, we recommend using a certified file format, as explained in [arXiv:2402.06083] and implemented in [https://github.com/fvigneron/Mandelbrot].曼德博集合（Mandelbrot set）是满足如下条件的参数c构成的集合：由z₀=0出发，按递推式z_{n+1}=z_n² + c生成的序列{z_n}始终有界。双曲分量（hyperbolic components）的中心是使得序列{z_n}为周期序列的参数c。这类中心按其周期分类：周期为满足z_k=0的最小正整数k，所有满足该条件的参数构成的集合记为Hyp(k)。 对于给定正整数n，由递推式p_n(c)=p_{n-1}²(c)+c（其中p₀(c)=0）定义的多项式p_n(c)的全部根，恰好是所有整除n的正整数k对应的周期k双曲中心。换言之，多项式p_n(c)的根集合，即为所有满足k|n的Hyp(k)的并集。 本数据集收录了曼德博集合中所有虚部非负的双曲中心Hyp(k)，涵盖周期k≤32的全部情况。每个文件包含若干根的列表，根以z=a+ib形式表示，其中a和b均被舍入至10^-26精度（即85比特精度）。文件采用逗号分隔值（CSV）格式，每行存储一个根，格式为字符串"a, b"，且注意b始终非负。对于周期26至32的文件，均采用BZ2算法进行压缩。 本数据集源自一份高精度且经过认证的数据库的十进制可读格式转换，相关细节参见参考文献[arXiv:2402.06083]。该完整数据库的覆盖范围可达周期41，若需获取其访问权限，请联系原作者。 若需获取预周期参数相关数据，请参考数据集[https://doi.org/10.5281/zenodo.15527762]。 注意事项：部分程序无法正确处理精度超出机器精度的浮点型CSV文件，本数据集的浮点值精度显然已超出该范围。为避免此类问题，建议以文本文件的形式通过编程方式读取CSV文件。为进一步保障工作流中的数据安全，推荐采用参考文献[arXiv:2402.06083]中所述并在[https://github.com/fvigneron/Mandelbrot]中实现的经认证文件格式。