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][1], M. Romera. Here is for example : [Calculation of the Structure of a Shrub in the Mandelbrot Set][2] 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][3]][3] Look also for: - [Sharkovskii's theorem][4] - [exponential map which transforms plane][5] [![Part of parameter plane with mini Mandelbrot sets for periods 1, 3, 9, 27, 81, 243. External rays are red.][6]][6] > "Many questions concerning (discrete) dynamical systems are of a number theoretic or combinatorial nature." Christian Krattenthaler HTH [1]: https://www.tic.itefi.csic.es/gerardo/Gframe.htm [2]: https://www.hindawi.com/journals/ddns/2011/837262/ [3]: https://i.sstatic.net/Vts8F.jpg [4]: https://en.wikipedia.org/wiki/Sharkovskii%27s_theorem [5]: https://commons.wikimedia.org/wiki/File:Feigenbaum_stretch_3.png [6]: https://i.sstatic.net/qKJjK.png