# 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:

Z_{i+1} = Z_{i}^{2} + c

if the iteration has period n, then Z_{0} is pre-periodic
if and only if:

Z_{0} = Z_{n}

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:

Z_{0} = 0.0000000000000

Z_{1} = -1.7903274919993

Z_{2} = 1.4149450366093

Z_{3} = 0.2117419646260

Z_{4} = -1.7454928324157

Z_{5} = 1.2564177360151

Z_{6} = -0.2117419646260

Z_{7} = -1.7454928324157 ≅ Z_{4}

Z_{8} = 1.2564177360151 ≅ Z_{5}

...

From this we see that the limit cycle has period 3, and consists
of the iterates Z_{4}, Z_{5}, and Z_{6}. Therefore, all iterates
before Z_{4} 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 "(CLR^{2})LRL". The preperiod is
the part in parentheses: C because Z_{0} is the critical point
(i.e. the origin), L for Z_{1} which is negative (i.e. to the left
of the origin), and R^{2} (shorthand for two R's in a row) because
Z_{2} and Z_{3} are both to the right of the origin; then we have
LRL which stands for the limit cycle points Z_{4}, Z_{5}, and Z_{6}
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) 1987-2023.

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