I have something similar I've used from time to time. It seems to give good results near deep minibrots. My implementation doesn't use fancy maths, though. It just does this:

1. Start with the center and period of the minibrot at the end of a zoom sequence/video.

2. Calculate exact scale and orientation of the minibrot, using the algorithm described elsewhere.

3. For each point being iterated, first affine transform it to the minibrot's coordinate space (i.e., apply an affine map that sends the minibrot center to 0, the minibrot spike tip to -2, etc.)

4. Iterate this point until a bailout radius is hit, often very large (e.g. 10^73).

5. At this point, transform the point by the inverse of the aforementioned affine transform, and also multiply the iteration count by the minibrot period.

6. Continue iterating until the "normal" bailout radius is hit.

7. Test for the point to be trapped during both phases. If the point is found to be trapped in the first phase, BIG iteration savings.

The tricky bit is the bailout in step 4. If it's too small (much below about 100) the curves of equal smoothed-iteration are not close enough to exactly circular at large distances, and you get seams. If it's too large you get seams or worse problems because it is sampling points from outside the period-doubling cascade around the minibrot (or from inside, but where it's not yet sufficiently close to perfectly rotationally symmetric) in essence.

The best bailout radius to use in step 4 is the power of ten equal to how much bigger than the minibrot is an image frame around the last off-center zoom on the way to that minibrot, OR the square of the size (relative to the minibrot) of the first exactly two-rotationally symmetric image of the "home stretch" after that point, OR if the minibrot is at the center of an embedded Julia the largest image that is inside the central "node" of that Julia: whichever is the farthest zoomed in of those three. I'd guess that the size of the atom domain for the mini might be a good guide -- in the embedded Julia case that domain tends to correspond closely to said central "node".

If the radius is too large in the off-center/2-fold-symmetric case, you get seams; too large in the embedded Julia case, you get Mandelbrot set dendrites or even minibrots in place of embedded Julia structures, because it renders chunks of the minibrot in whose "field" the Julia was found in lieu of that Julia.

The other failure mode is that shapes can be wrong if they are being shape-stacked, or are embedded Julias. The first case happens if e.g. you take a square shape, zoom repeatedly at one corner of the square, then zoom to a period-doubled field about a mini to find a bow tie shape, then zoom into that deeper. If the bailout disk around the central mini is in the center of the bow tie you're fine, but if it includes the ends of the bow tie, the secondary bow ties in the centers of the ends will be squares instead of bow ties, and recursive similar errors will occur at greater depths in the sequence. This is only a concern if the secondary bow ties in the centers of the ends are big enough to be visible in an image that shows the whole bow tie. With a shallow square this is a concern, but with a deep square it can't happen as the next level of bow ties will be smaller than a pixel at any sane rendering resolution. The analogous failure will happen from stretching and conjoining other shapes than squares; I used square->bow tie as just an example.

Where the "true" minibrot has embedded Julias in the field around it if you zoom deeper into twofold areas near the mini, with this method you will get copies of that minibrot itself in place of the Julias. So a) the technique can't be used at all around a mini shallow enough that the embedded Julias around it are bigger than a pixel in an image showing the full mini; b) when zooming *past* the mini it can only be used down as far as when an embedded Julia associated to that mini would be close to coming into view (becoming bigger than 1/100 pixel or so). Past that point, deviations in the shapes of the dwell bands around the Julia shape vs. around a Mandelbrot shape can become macroscopic (greater than 1 pixel deviation size).

Needless to say, a mini that is noticeably asymmetrical can't be used as the target. Minis much shallower than the transition between double and arbitrary precision cannot, in general, be used. The deviation between the transformation that maps the mini onto the main set and an affine approximation (first, linear term in your Taylor series, essentially) cannot be sufficient to make distortions bigger than a pixel in size or it won't work (you'll get seams again in that case).

This method has been tested and, with the limitations noted above, works reasonably well to massively speed up rendering about deep minis, especially views where the mini's interior occupies multiple image pixels. There are multiple contributing factors. Let's consider a hypothetical example of a deep mini at 1e200 with the last off-center zoom around 1e100, from deep structure stacking rather than an embedded Julia center; its period is 4000. The image around the last off-center zoom has iterations averaging 20,000.

1. If deeper images about this center are rendered without this technique, the iterations range from 20,000 to 24,000 down to the 3/4 mark; 24,000 to 28,000 down to the 7/8 mark; in the millions once near the minibrot; and you'll need billions to render the minibrot border accurately. With this speedup, if you want the minibrot border rendered as accurately as without it with 4 billion iterations as max, the iterations done in phase 1 range from 1 up to one million, and the iterations done in phase 2 range up to about 20,000. Except right at the end of the sequence when the minibrot itself occupies a positive number of pixels, the number of first-phase iterations tends to be no more than a handful.

2. The first set of iterations can be done with doubles rather than arbitrary precision. And is the potentially most numerous. Unless you zoom far enough *past* the target minibrot that zooming that far past the main set would require arbitrary precision.

3. The second set of iterations can be done with half the bit precision (and thus 1/4 the multiply times) as to do a naive render job on a closeup of the minibrot. The precision of these iterations over the whole zoom sequence rises to this point at the halfway mark, then stops increasing instead of continuing to increase linearly over the "back half".

4. The second set can be done using perturbation theory, so the two giant speedups can be combined.

Note that item 3 alone reduces the time for the deepest images to 1/4 what it would be. Items 1 and 2, though, really help: item 1 is already giving its own fourfold speedup by the time you're at 16-fold symmetry, and the combination makes rendering a very accurate image of the deep minibrot with very crisp edges and no "black mud leaking out" take minutes (perturbation) or weeks (no perturbation) instead of days (perturbation) or years (no perturbation).

One more caveat: zooming *past* the target mini and then switching this *off* again to zoom to an even deeper mini runs into a new problem: though the shapes in each individual frame of the sequence will be accurate, there's a "drift" caused by the affine transform of mini to full set being an approximation to a more complex mapping. As a result going deep enough past the mini and then switching this speedup off will result in being "teleported", potentially, some distance through the shapes in the dwell bands around the mini. The best way to avoid this is likely to be to a) switch it off as soon as feasible, e.g. as soon as no more minibrot interior is in the frame, and b) if necessary jump the center coordinates at that same point so as to recenter on the correct structure.