In this paper, we research more in depth properties of Backtracking New Q-Newton's method (recently designed by the third author), when used to find roots of meromorphic functions. If $f=P/Q$, where $P$ and $Q$ are polynomials in 1 complex variable z with $deg (P)>deg (Q)$, we show the existence of an exceptional set $\mathcal{E}\subset\mathbf{C}$, which is contained in a countable union of real analytic curves in $\mathbf{R}^2=\mathbf{C}$, so that the following statements A and B hold. Here, $\{z_n\}$ is the sequence constructed by BNQN with an initial point $z_0$, not a pole of $f$. A) If $z_0\in\mathbb{C}\backslash\mathcal{E}$, then $\{z_n\}$ converges to a root of $f$. B) If $z_0\in \mathcal{E}$, then $\{z_n\}$ converges to a critical point - but not a root - of $f$. Experiments seem to indicate that in general, even when $f$ is a polynomial, the set $\mathcal{E}$ is not contained in a finite union of real analytic curves. We provide further results relevant to whether locally $\mathcal{E}$ is contained in a finite number of real analytic curves. A similar result holds for general meromorphic functions. Moreover, unlike previous work, here we do not require that the parameters of BNQN are random, or that the meromorphic function $f$ is generic. Based on the theoretical results, we explain (both rigorously and heuristically) of what observed in experiments with BNQN, in previous works by the authors. The dynamics of BNQN seems also to have some similarities (and differences) to the classical Leau-Fatou's flowers.

翻译：本文深入研究了回溯新Q-牛顿法（由第三作者近期提出）在求解亚纯函数根时的性质。若 $f=P/Q$，其中 $P$ 和 $Q$ 为单复变量 $z$ 的多项式且满足 $deg (P)>deg (Q)$，我们证明存在一个例外集 $\mathcal{E}\subset\mathbf{C}$，该集合包含于 $\mathbf{R}^2=\mathbf{C}$ 中可数个实解析曲线的并集，使得以下命题A与B成立。此处 $\{z_n\}$ 表示以非 $f$ 极点的初始点 $z_0$ 通过BNQN构造的序列。A) 若 $z_0\in\mathbb{C}\backslash\mathcal{E}$，则 $\{z_n\}$ 收敛至 $f$ 的某个根。B) 若 $z_0\in \mathcal{E}$，则 $\{z_n\}$ 收敛至 $f$ 的某个临界点——但非其根。实验结果表明，即使 $f$ 为多项式，$\mathcal{E}$ 集通常也不包含于有限个实解析曲线的并集中。我们进一步探讨了 $\mathcal{E}$ 在局部范围内是否包含于有限条实解析曲线的问题。类似结论对一般亚纯函数同样成立。此外，与先前研究不同，本文不要求BNQN的参数具有随机性，亦不要求亚纯函数 $f$ 具有一般性。基于理论结果，我们（从严格论证与启发式角度）解释了作者前期工作中BNQN实验所观测到的现象。BNQN的动力学行为与经典Leau-Fatou花瓣结构既存在相似性，也存在差异性。