Where $f(z)=z^2+c$ is the Mandelbrot iteration function, are there any known complex numbers $z$ such that iterating $z\to f(z)$ to infinity retains $z$ on the boundary (i.e. it does not explode to infinity or collapse to zero)?
If such known points exist, are they periodic?
Can we prove that they do or don't exist, without knowing their values?
Misiurewicz points lay on the boundary of the mandelbrot set and become periodic after some $n$ iterations. Branches of the set are proved to be these points, but their exact values are unknown, only approximated.