Pre-Periodic Point
Robert P. Munafo, 2023 Mar 22.
A pre-periodic iterate is a value of Z that is not in the limit cycle for the value of c being iterated. Given the recurrence relation:
Zi+1 = Zi2 + c
if the iteration has period n, then Z0 is pre-periodic if and only if:
Z0 = Zn
Misiurewicz points are pre-periodic points. For example, the tip of R2F(1/2B1)S is at about -1.790327491999346 + 0i. The first 8 iterates (rounded off a bit) are:
Z0 = 0.0000000000000
Z1 = -1.7903274919993
Z2 = 1.4149450366093
Z3 = 0.2117419646260
Z4 = -1.7454928324157
Z5 = 1.2564177360151
Z6 = -0.2117419646260
Z7 = -1.7454928324157 ≅ Z4
Z8 = 1.2564177360151 ≅ Z5
...
From this we see that the limit cycle has period 3, and consists of the iterates Z4, Z5, and Z6. Therefore, all iterates before Z4 are preperiodic.
Since this Misiurewicz point is on the real axis we can derive its CLR-style name as used by Romera, Pastor, and Montoya in e.g. their 1996 paper "On the cusp and the tip of a midget in the Mandelbrot set antenna". They refer to it as "(CLR2)LRL". The preperiod is the part in parentheses: C because Z0 is the critical point (i.e. the origin), L for Z1 which is negative (i.e. to the left of the origin), and R2 (shorthand for two R's in a row) because Z2 and Z3 are both to the right of the origin; then we have LRL which stands for the limit cycle points Z4, Z5, and Z6 which are negative, positive, and negative respectively.
revisions: 20020527 oldest on record; 20230322 add example using the iterates of R2F(1/2B1)St
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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