Robert P. Munafo, 2023 Mar 27.
The line containing the points on the complex plane that are real numbers. It is normally oriented horizontally with the negative numbers on the left.
The Mandelbrot set on the real axis is relatively easier to study, inasmuch as all calculations needed are in real numbers, and there are useful constraints such as the fact that any range of values only has three possible relationships to some other range: above, below, or intersecting.
The behaviour of real-valued iterated functions has been heavily studied. For our purposes the functions of interest are those that map a fixed-size range onto the same range twice, with one part decreasing and the other increasing. In 1973 Metropolis et al. give in detail the method to find the ordering and periods of all islands on the real axis; see MSS Algorithm for a description.
Quite a lot of work has been done by Romera et al. to illustrate the structure of the Mandelbrot set using the real axis (and particularly the spike) as an example. They describe the filaments, Misiurewicz points, and islands along with their external angles, etc..
revisions: 20080228 oldest on record; 20230327 reference "the spike", Metropolis, MSS algorithm, Romera, etc.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Mar 28. s.27