In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm): $$ f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$ with the question whether, and if: to what, iterations converge if started at, say, $x_0=3$.
Let for convenience $\beta$ denote $\beta = \log(b)$ .
I find empirically, that for $\beta \ge 1$ $$ \lim_{h \to \infty} f^{\circ h}(x_0) = 1 $$ where the convergence is monotonuously from above. For some $\eta_1 \approx 0.498 \lt \beta \lt 1$ the limit is still $1$ but it converges from above and below $1$
For a range of smaller bases $b=\exp(\beta)$ with $\eta_2 \approx 0.3999 \lt \beta \lt 0.498 \approx \eta_1 $ we get convergences to sets of accumulation points, with set lengthes $12,24,48,96, ??? $ which might possibly extend to multiples of $12$ (or even of $3$) with arbitrary powers of $2$ .
Finally, for $ \beta \lt \eta_2$ it seems, that there is no more convergence, and the sequence of iterates produces numbers which look quite random.
Q1: Can we approximate the values of the $\eta$'s more precise? Are there analytical descriptions for them?
Q2: Are the possible set-lengthes for the accumulation points indeed $12 \cdot 2^k$ ?
Examples: For $\beta = 0.41$ ,$b \approx 1.50681778511 $ I get after 1000 initial iterations the following sequence of further iterates (read along rows)
...
0.264550328299, 3.50777903235, 0.446844683843, 2.41158665503,
0.264550328299, 3.50777903235, 0.446844683843, 2.41158665503,
0.264550328299, 3.50777903235, 0.446844683843, 2.41158665503,
0.264550328299, 3.50777903235, 0.446844683843, 2.41158665503,
0.264550328299, 3.50777903235, 0.446844683843, 2.41158665503,
0.264550328299, 3.50777903235, 0.446844683843, 2.41158665503,
0.264550328299, 3.50777903235, 0.446844683843, 2.41158665503,
0.264550328299, 3.50777903235, 0.446844683843, 2.41158665503,
...
The four different values are what I call "accumulation points" and the set-length for this base is just 4 .
[update] Although the question Q2 has been resolved/handled - here is the table to which I have referred when I was talking about sets of accumulation-points. However I had not noticed, that the cardinalities of the sets of accumulation-points are also depending on the starting value $x_0$ besides the parameter for the exponential base $b \lt \sqrt e $ and can be arbitrarily created. Initial value for this table was in all cases $x_0=3$: $$ \begin{array}{l|l} \log(b) & \text{"set-length" or } \\ & \small \text{"No. of accumulation-points"} \\ \hline\\ 0.4 & 12 \\ 0.3999 & 24 \\ 0.39988 & 24 \\ 0.39987 & 24 \\ 0.399865 & 48 \\ 0.39986 & 48 \\ 0.399858 & 96 \\ 0.399857 & 96\\ 0.39985 & \small \text{-no sufficient convergence } \\ & \small \text{ achieved in 100000 iterations-}\\ \end{array}$$
One important point in the system is $b=\sqrt{e}$, as was pointed out by Gottfried. For $b>\sqrt{e}$, there is a stable attracting fixed point for $f(x)=x-\log_b(x)$, and that fixed point is x=1. For $1.518120456732599974768513856<b<\sqrt{e}$, the fixed point in the neighborhood of one is repelling, and iterates starting in the neighborhood of one (but not equal to one) settle into a two cycle orbit. This value, $\approx1.5181$ is the next critical point in iterating f(x), and for bases less than that value, the system once again bifurcates.
For $1.499042192220287185464351750<b<1.518120456732599974768513856$, the fixed point starting in the neighborhood of one settles into a four cycle orbit. Below that point, the system would settle into an eight cycle orbit. I didn't calculate the next bifurcation point, where the eight cycle orbits become sixteen cycle orbits. As I understand complex dynamics, there is an infinite sequence of these power of two bifurcations, with each region smaller than the previous bifurcation region. This is Mandelbrot like behavior, though I wouldn't presume to know how to make a "Mandelbrot" like complex graph for Gottfried's function.
Gottfried looked at $b=\exp(0.4)$, which has a twelve cycle fixed point. This is a region of stability, that is actually past the infinite sequence of bifurcations, akin to a mini-Mandelbrot in the Mandelbrot set. Before getting to the mini-Mandelbrot region, you have to get past the infinite sequence of bifurcations, where chaos occurs, which seems to be near $b=\exp(0.4015293)$. For example, $b=\exp(0.4015295)$ is in the 512-cycle region, which is very close to the chaotic boundary.
edited with images updated. See this answer, How to figure out the starting point for this Mandelbrot? which has the ideal $z_0=1/\log(b)$ as the starting point for iterating f(x). Here is the main Mandelbrot "bug" generated from Gottfried's iterated function. Gridlines for the Mandelbrot image are 1/10, with the function varying from 1.425 to 1.725. You can see the main bifurcation line at $\exp(0.5)\approx1.65$. It looks like stackexchange resized the image, but if you right click, you can view the original at 750x500.
The algorithm I used works pretty well, with the $z_0$ starting point. Unfortunately, it appears that stackexchange edited out the exif comments from the .jpg files. Here is a zoom in, from 1.491 to 1.519, with grid lines of 1/100, showing the 8x bifurcation region. On the left, you can just make out Gottfried's region at 1.4918.
Here is the tip, from 1.44 to 1.50, with grid lines of 1/100. You can see several of the mini-mandelbrots.
Here is the biggest mini-mandelbrot, from 1.452 to 1.4544, with grid lines of 1/1000. This mini-mandelbrot has a main bulb with a 3-cycle, as compared with the 12-cycle from the mini-mandelbrot in Gottfried's question.
Finally, here is the amazing wide view, showing the infinite spiral, of radius $\exp(0.5)$, with the real values ranging from +/-1.66, and the imaginary maximum at 1.66 as well. Everywhere outside of this infinite circular spiral, the fixed attracting point is 1. The very larger black region in the center of the spiral between -0.6 and +1 is a computation artifact, where the base is close to zero. This is because the function escapes to +infinity instead of -infinity, so the algorithm incorrectly regards this as a stable fixed point region.
Here is a link to the pari-gp code. http://www.sheltx.com/pari/gottfried.gp