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Newton-Raphson Zooming

Robert P. Munafo, 2023 Jun 20.

This is a method for automatically zooming most of the way into a 2-fold embedded Julia set, or its central island, given a center and size that is suspected to contain one. This is essentially just the Newton-Raphson method (see that article) for finding an island followed by recentering and some amount of zooming.

This method builds on a simpler technique that I'll describe first:

• Perform the Jordan curve method to determine the period of the lowest-period island (also described in the period article) and to locate its approximate position
• Adjust the center and decrease the size (i.e. zoom in)
• Pepeat until the size is close to that of the 2-fold embedded Julia set (which is not automatic, but a person can watch the screen and stop the zooming at the right time, or a program can automatically snapshot the image with each 10-fold increase in magnification, for later examination by the user.)

Newton-Raphson zooming differs by using Newton's method to find the location of the island, rather than inverse interpolation for a quadrilateral (as suggested in the Jordan curve method article). This is better for two reasons:

• the island nucleus position is exact to within the precision being used, rather than being just a first approximation; this allows zooming in by bigger steps (such as all the way in to the island itself)
• the Newton iteration involves computing the derivative of each iteration, and it takes just a few more operations to optain a size estimate for the island, see stability window for details. The additional operations include a series sum of the reciprocals of partial products of the first p-1 iterates (where p is the period found in the first step)

Size Estimation of Embedded Julia Sets

When the size s of the island has been estimated, Claude Heiland-Allen found that the size of its 2-fold embedded Julia set is approximately s3/4, although he also stated uncertainty regarding whether that works all the time.

I have found that the outermost 2-fold form of a second-order embedded Julia set occurs twice as deep as the island at the center of the first-order embedded Julia set that was encountered on the way, which in turn is twice as deep as its 2-fold first-order embedded Julia set. In such a case, given the size estimate s of the second-order embedded Julia set's island, the size of the 2-fold second-order embedded Julia set would be s1/2; whereas s3/4 would be the size estimate of the 4-fold embedded Julia set that is found on the way from the 2-fold one to its island.

I suspect that Claude Heiland-Allen's experience comes mostly from Leavitt navigation, which involves periodically redireting the zoom towards paramecia (sparse Fatou-like embedded Julia sets, also called peanuts).

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Jun 27. s.27