If you change the form of your map to $z_{n+1} = z_n^2 + c $ ( conjugation) and take only real values of c ( real slice of Mandelbrot set) then you will find the answers in papers by G. Pastor, M. Romera. Here is for example : [Calculation of the Structure of a Shrub in the Mandelbrot Set][1]

The part from 0 to Feigenbaum point in your map is a periodic region where period doubling cascade occurs. In c plane it is from 0.25 to F

The part from F to 4 in your map is a Mandelbrot set antenna. It's structure is described in that paper : 

[![IMage from paper][2]][2]

Look also for [Sharkovskii's theorem][3]

HTH


  [1]: https://www.hindawi.com/journals/ddns/2011/837262/
  [2]: https://i.sstatic.net/Vts8F.jpg
  [3]: https://en.wikipedia.org/wiki/Sharkovskii%27s_theorem