Root
Robert P. Munafo, 2010 Sep 9.
The unique point on the boundary of a mu-atom of period P where two external angles of denominator 2P-1 meet.
All mu-atoms have a root; if the mu-atom is a cardioid then the root is its cusp; if the mu-atom is a child then the root is the bond point where the mu-atom touches its parent.
Some examples:
The root of R2a is 0.25+0i (the Elephant valley cusp R2.C(0)).
The root of R2.1/2a is -0.75+0i (the Seahorse valley cusp R2.C(1/2)).
The root of R2F(1/2B1)Sa is -1.25+0i (the cusp R2F(1/2B1)S.C(0)).
The root of R2F(1/3B1)S is a cusp at approximately -0.154724+1.031047i.
The term appears to originate with comparisons of a mu-unit to the graph theory concept of "tree". A typical usage is in Jung (in the 2003 version of "Homeomorphisms of the Mandelbrot Set"), page 6:
Parameter rays RM(θ) are defined as preimages of straight
rays [...] When θ is periodic, then c is the root of a hyperbolic
component (see below)
[...]
Hyperbolic components of M consist of parameters, such that the
corresponding polynomial [(referring to Z2+C)] has an attracting
cycle. The root is the parameter on the boundary, such that the cycle
has multiplier 1. The boundary of a hyperbolic component contains a dense
set of roots of satellite components. [...]
Another example is from Milnor's 1999 paper "Periodic Orbits, Externals Rays and the Mandelbrot Set ..." in the figure 12. This shows R2F(1/3B1)S with external rays leading to the island's cusp and to the bond point where its cardioid R2F(1/3B1)Sa touches its north bulb R2F(1/3B1)S.1/3a. The 3/15 and 4/15 external arguments meet at the cusp, and the 820/4095 and 835/4095 external arguments meet at the bond point. The caption reads:
Detail of the Mandelbrot boundary, showing the rays landing at the root points of a primitive period 4 component and a satellite period 12 component.
In the above quotes, the authors use the term hyperbolic component the same way Mandelbrot does; satellite is what I call a child.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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