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I'm recreationally learning about fractals, specifically about functions of the "Multibrot" type $f(z) = z^d + c$. Just recently I became aware of the fact that repelling periodic points are just a vanishingly small subset of a Julia set. Thus, my initial intuition, that each point in the set must be part of some (arbitrarily and thus possibly, infeasibly huge) periodic cycle, was totally wrong.

Now the most interesting piece of information I have found is in this answer -- it sketches how the binary expansions of the starting point can lead to different dynamics, including the periodic as well as non-periodic case.

I would very much like to learn more about how the binary expansions are related to the corresponding orbits, but there are no references provided. In particular, the connections of "bit-shifted" binary expansions and periodic orbits, and what is known about properties of the non-periodic, irrational case (similar to the 0.010010001.... example), which appears to be the answer to my question about "Julia set points that are not periodic".

I assume that this is from some book or paper where I could learn more? Maybe this is well-known in the field, but I don't know what exactly to search for. Before I ask in a comment in a decade-old question, I thought I rather create a new one. Thanks a lot in advance!

apirogov
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  • for $z \to z^d + c$ the expansion in base $d$ is relevant because $z^d$ on the unit circle is like $d \theta \pmod{1}$ on angles measured in turns – Claude Nov 27 '23 at 16:30
  • This is concerning base d expression for the angle, i.e. when working in polar coordinates, right? 2. I noticed that I can reach a specific repelling periodic cycle using inverse iteration using a kind of d-ary Lyndon word to choose roots along the orbit out of the multiple solutions. Do you happen to know how this is (possibly) related?
    • – apirogov Nov 28 '23 at 10:31
  • it's probably related to kneading sequences or external rays. I think a more robust way to find a specific preimage is to keep track of $P$ points, starting far away at the angles you want ($P$-periodic under $\theta \to d \theta \pmod{1}$) and pick the root that's nearest to the point corresponding to the angular preimage each time, IIRC it is proven that this chooses the correct root at each step (at least for $d=2$), but I don't have a reference handy... – Claude Nov 28 '23 at 10:46
  • Thanks a lot for the hints, all that sounds very relevant! What I'm doing sounds a bit similar - I use a "dynamic" numbering of roots, starting with the closest phase as the "principal"/0-th root, and given a e.g. ternary (d=3) sequence such as 0121 I pick 0,1,2,1,0,1,2,1,..th root, which will find me the corresponding period 4 cycle. The initial point does not seem to matter, but I stick to 0 as starting point. It works quite reliably for parameters where the Julia set is not connected. This is all empirical experimentation, wish I could read up on what is going on and when+why it works. – apirogov Nov 28 '23 at 12:32