Petals for points near the origin 
Let $P(z)=-z+z^{p+1}$ where $p$ is a positive integer.  How many petals does $P$ have at the origin?

From a few examples in my notes, I noticed the following:


*

*$-z+z^{14}$ has 13 petals. 

*$-z+z^3$ has 4 petals.


I'm not able to find a proof of the result above, but I am able to deduce that there are:


*

*an $2p$ petals if $p+1$ is odd 

*if $p+1$ is even then there are $p$ petals.  


Could you provide some intuition, or a sketch of the result?
There is a theorem I'm leaning on to get some information:

Let $R$ be a rational map and suppose that $R(z)=z+az^{p+1}+...$ near the origin.  Let $\Pi_j$ be the petals of $R$ and for each $j$ let $F_j$ be the component of the Fatou set that contains $\Pi_j$.  Then $R^n(z) \to 0$ and $\arg R^n(z) \to 2\pi k/p$ on $F_k$ and $F_0,...,F_{p-1}$  are distinct components of the Fatou set.

But using this theorem, it only shows me that there should always be $p$ petals (ex, there is a petal in between every $\frac{2\pi k_i}{13}, \frac{2\pi k_{i+1}}{13}$).  
Besides getting a sketch of the proof, I have one more questions:


*

*In the even case, there is an overlap between attracting or repelling petals?

 A: 
Let $P(z)=z^{p+1} - z$ where $p$ is a positive integer.  How many petals does $P$ have at the origin?

In the parabolic case critical orbit looks like n-th arm star
Arms tend to attracting directions near fixed point. Number of attracting petals is equal to n. 
So I have checked critical orbits using this Maxima CAS script. Here are examples for p=2 and p=13


My results are listed below:   


*

*p = 1  ( degree =  2) there are  2 petals

*p = 2  ( degree =  3) there are  2 petals

*p = 3  ( degree =  4) there are  6 petals

*p = 4  ( degree =  5) there are  4 petals

*p = 5  ( degree =  6) there are 10 petals

*p = 6  ( degree =  7) there are  6 petals

*p = 7  ( degree =  8) there are 14 petals

*p = 8  ( degree =  9) there are  8 petals

*p = 9  ( degree = 10) there are 18 petals 

*p = 10 ( degree = 11) there are 10 petals

*p = 13 ( degree = 14) there are 26 petals


so:


*

*for p odd there are 2*p petals

*for p even there are p petals

A: 
In the even case, there is an overlap between attracting or repelling petals?

It depends what definition of petal you use.  There is no unified definition of petals.
Petal of a flower can be :


*

*attracting / repelling

*small/alfa/big/ 


Small attracting petals do not ovelap with repelling petals, but big do.  
One can choose overlapping ( red) or not ( yellow)  petals 
