Boundary about the Mandelbrot Set Let $f_c : z \mapsto z^2+c$ and the sequence $\displaystyle \left(z_n\right)_{n \in \mathbb{N}}$ defined by $z_0=0$ and $z_{n+1}=f_c\left(z_n\right)$.
We say that $c \in M$ if the sequence $\displaystyle \left(z_n\right)_{n \in \mathbb{N}}$ remains bounded.
I want to prove that if it exists $p$ such that $z_{n+p}=z_n$ starting at $n_0>0$ then $c$ relies in the boundary of $M$. Is there a way to show this ?

If $c=0$ then $z_n=0$ for all $n\in\mathbb{N}$, but $0$ does not belong to the boundary of the Mandelbrot set. Actually $0$ is in the interior part of the Mandelbrot set.
Probably you mean the definition of Misiurewicz point $c$ for which the orbit of $0$ is strictly pre-periodic. In that case the property is true. A proof can be found in Etudes dynamique des polynômes complexes (english version), Chapter 8,
by Adrien Douady and John H. Hubbard. It is based on the following deep result: there is a conformal isomorphism from the exterior of the unit disk to the complement of the Mandelbrot set.
