My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, $f(z)\mapsto z-\log_b(z)$. In particular, for each base, there's a well defined Julia, and if $\Re(z)<-1$, then we can say the iterates of f(z) get arbitrarily large negative. For other points we observe f(z) iterates towards a stable attracting cyclic orbit. The first bifurcation occurs at $b=\exp(0.5)$. For $\Re(b)>\exp(0.5)$, z=1 is an attracting fixed point, and for For $\Re(b)<\exp(0.5)$, z=1 is a repelling fixed point.
Then I wondered whether a Mandelbrot plot could be generated as the logarithmic base b varies. I got as far as generating a Mandelbrot plot for the escape iteration count, iterating of f(z), but I'm having some serious difficulties. I'm still puzzled as to the correct algorithm to generate these Mandelbrots, because the required starting point seems to be a function of b, the logarithmic base. Is there an algorithm to calculate a good starting point for iterating the Mandelbrot $f(z)\mapsto z-\log_b(z)$, such that the starting point is guaranteed to be in the cyclic attracting basin for b, if base(b) has an attracting cycle?
For the normal Mandelbrot, $f(z)\mapsto z^2+c$, the starting point used for each value of "c" is z=0, which is the center of the corresponding Julia for c. The exterior of the Julia can be put into correspondence with a Botcher function for z^2, for $|z|>1$. Bonus question: Is there a corresponding Botcher function for Julias for Gottfried's f(z) function, and can these Julias be put into correspondence with the Julia's for the normal Mandelbrot set? Gottfried's Julia's are not symmetrical, and actually have infinitely large positive points that iterate into the stable attracting basin.
For example, there are flaws in this Mandelbrot plot, where some of the points that are colored really belong to a cyclic basin, and instead should be black since the cyclic basin never escapes. Here, I started iterating with z=2.6, which is an ok starting point for this plot, but its not perfect. I tried other plots, where I use multiple starting points, which is a little better, but far from ideal. This plot varies from b=1.425 to b=1.725 with grid lines of 1/10th.