**Shrubs ordering**

*Miguel Romera*
As can be seen in [1], the end of the period-doubling cascade of any
hyperbolic component *hc* is its Myrberg-Feigenbaum point
**MF**(*hc*). What emerges from **MF**(*hc*) is what we
denominate the *shrub* of the *hc*, due to its shape. If the shrub
emerges from a primary *hc (p/q)*, then we have a *primary shrub*,
the shrub*(p/q)*. If the shrub emerges from a secondary, tertiary,...,
N-ary *hc*, then we have a secondary, tertiary ,..., N-ary shrub.
Secondary, tertiary, ..., N-ary shrubs have 2, 3, ..., N subshrubs. As can be
seen in [1], we can calculate the period of the representative hyperbolic
component *rhc* of each one of the structural branches of the shrubs,
as well as the preperiod and period of each one of the Misiurewicz points
from which structural branches emanate (nodes) or where the shrub branches
end (tips).

# Primary shrubs ordering

1. Shrub(*1/3*)
Shrub(*1/3*) is associated to a "decoration" formed by a *node* that
is a Misiurewicz point from which three (*p = 3*) structural *branches*
emanate. We can consider that the period-3 hyperbolic component *1/3*,
the origin of the shrub(*1/3*), is placed on the main branch *0*,
and that this branch *0* finishes on the main node *0*, a Misiurewicz
point *M*_{4,1}. If we turn clockwise around the main node by
following the branches 1 by 1 (because *q = 1*), we have the branches
*0, 1* and *2* (or the *hc*'s with periods *3, 4* and
*5*). At the end of each branch with a period-p *rhc* we have the
Misiurewicz point *M*_{p+1,1}. So, at the end of the branch *1*,
whose *rhc* has period *4*, we have the Misiurewicz point
*M*_{5,1}. If we turn clockwise around the *M*_{5,1}
by following the branches 1 by 1, we have the *hc*'s with periods *4,
5* and *6*. Tips can also be found, being *ftip(1/3) =
M*_{5,1}. In such a manner we can find all the structural *hc*'s
and Misiurewicz points.

*Comments* As we know, the Mandelbrot set is similar locally to the
corresponding Julia set. Therefore one could explore quadratic maps on
dynamical plane (especially unstable periodic orbits ordering) to explain
the M-set shrubs too.
2. Shrubs(*2/5*)

Shrubs(*2/5*) is associated to a "decoration" formed by a *node*
from which five (*p = 5*) structural *branches* emanate. If we turn
clockwise around the main node (but now by following the branches 2 by 2!
because here *q = 2*) we have the branches *0, 1, 2, 3* and *4*
(or the *hc*'s with periods *5, 6, 7, 8* and *9*). Again, at
the end of each branch with a period-p *rhc* we have the Misiurewicz
point *M*_{p+1,1}. For example, at the end of the branch *2*,
whose *rhc* has period *7*, we have the Misiurewicz point
*M*_{8,1}. If we turn clockwise around this *M*_{8,1}
by following the branches 2 by 2, we have the hyperbolic components with
periods *7, 8, 9, 10* and *11*. Here *flip(2/5) = M*_{5,1}.
In such a manner we can find again all the structural *hc*'s and
Misiurewicz points.
# Secondary shrubs ordering

Shrub(*1/3 1/5*) has two subshrubs, subshrub_{1}(*1/3 1/5*)
and subshrub_{2}(*1/3 1/5*), that are associated to "decorations"
formed by a *node* from which five (*p*_{2} = 5) and three
(*p*_{1} = 3) structural *branches* emanate. In both
subshrubs we turn clockwise around nodes by following the branches 1 by 1
because in this case *q*_{1} = q_{2} = 1. However, in
the first subshrub periods increase 3 by 3 whilst in the second subshrub
periods increase 1 by 1 [1]. At the end of each branch of period *p*
we have the Misiurewicz point *M*_{p+1,3} in
subshrub_{1}(*1/3 1/5*) and *M*_{p+1,1} in
subshrub_{2}(*1/3 1/5*). So, at the end of the branch *2*
of subshrub_{1}(*1/3 1/5*), whose *rhc* has period *21*,
we have the Misiurewicz point *M*_{22,3}. If we turn clockwise
around this *M*_{22,3} by following the branches 1 by 1, we have
the *hc*'s with periods *21, 24, 27, 30* and *33*. Likewise, at
the end of the branch *1*^{-} -> 2 of the
subshrub_{2}(*1/3 1/5*), whose *rhc* has period *14*,
we have the Misiurewicz point *M*_{15,1}. If we turn clockwise
around the *M*_{15,1} by following the branches 1 by 1, we have
the *hc*'s with periods *14, 15* and *16*. In such a manner we
can find all the structural *hc*'s and Misiurewicz points in both
subshrubs. See [1] to calculate tips in both shrubs.

Choose another shrub to "remove" labels.
Open in a new window The Julia sets trip with
orbit to determine *hc*'s periods.
[1] *M.Romera, G.Pastor, G.Alvarez, and F.Montoya,*
"Shrubs in the Mandelbrot Set Ordering",
accepted in International Journal of Bifucation and Chaos, 2002.

Contents
Previous: Reverse period n-tuplings cascades

*updated* 25 March 2003