Determine whether a point $c$ is in a wake $W_P$.

Question

One can draw wakes in the parameter plane of the Mandelbrot set, by tracing external rays inwards from near $\infty$ (with Newton's method or other algorithm) to get polygonal outlines which can be filled. But reading the following paper I think there should be a way to plot them implicitly, starting from each pixel's $c$ coordinate to a binary classification:

Definition. Let $O = \{z_1, \ldots, z_p\}$ be a periodic orbit for $f$ . Suppose that there is some rational angle $t \in \mathbb{Q}/\mathbb{Z}$ so that the dynamic ray $R^{K(f)}_t$ lands at a point of $O$. Then for each $z_i \in O$ the collection $A_i$ consisting of all angles of dynamic rays which land at the point $z_i$ is a finite and non-vacuous subset of $\mathbb{Q}/\mathbb{Z}$. The collection $\{A_1, \ldots, A_p \}$ will be called the orbit portrait $P = P(O)$.

Theorem 1.2. The Wake $W_P$. The two corresponding parameter rays $R^M_{t\pm}$ land at a single point $r_P$ of the parameter plane. These rays, together with their landing point, cut the plane into two open subsets $W_P$ and $\mathbb{C} - W_P$ with the following property: A quadratic map $f_c(z) = z^2 + c$ has a repelling orbit with portrait $P$ if and only if $c \in W_P$, and has a parabolic orbit with portrait $P$ if and only if $c = r_P$.

-- Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account, John W. Milnor https://arxiv.org/abs/math/9905169

However, I haven't yet deciphered if the paper provides a constructive algorithm for checking, or if it is just an existence proof.

Question: given an orbit portrait $P$, provide a constructive algorithm to determine if a point $c \in \mathbb{C}$ is in the wake $W_P$.

Answer 1

One way of drawing them implicitly is by tracing the $2$ rays corresponding to the wake's characteristic angles in the dynamical plane and seeing if they land together, for each pixel $c$ of the parameter plane:

Theorem 2.5 (Ray portraits are born in wakes). Every cycle of periodic dynamic rays has two characteristic addresses/angles $s_+ , s_−$ ; the parameter rays of addresses/angles $s_+ , s_−$ land together at a parabolic parameter and hence partition the plane into two regions, one of which does not contain the unique period one hyperbolic component and which is called a parabolic wake $W_{s_− ,s_+}$. The two dynamic rays of address/angle $s_+ , s_−$ land together for a map $f_c \in F$ if and only if $c \in W_{s_− ,s_+}$.

-- A survey on MLC, Rigidity and related topics, Anna Miriam Benini https://arxiv.org/abs/1709.09869

The boundaries are a bit rough (I don't know why), and convergence near the parabolic point is slow (which I expected), but it seems to give reasonable images in a brief test: