Backtracking New Q-Newton's method for finding roots of meromorphic functions in 1 complex variable: Global convergence, and local stable/unstable curves

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.