Fractal behavior of tetration

[bo198214]

I suppose the Julia set of exponentiation would be the convergence region of \( {}^{\infty}z \), right?

By Wikipedia the Julia set of an entire function is

the boundary of the set of points which converge to infinity under iteration

For \( \exp_b \) this would be the boundary of all points \( x \) with \( \lim_{n\to\infty} {\exp_b}^{\circ n}(x)=\infty \).

I think for \( b>e^{1/e} \) this is the whole complex plane, because the points that go to infinity are next to points that doent.

For tetration the Julia set are is the boundary of points \( x \) such that \( \text{sexp}_b^{\circ n}(x)\to\infty \). I guess this depends on which tetration we choose. So maybe Dmitrii can draw a picture .

Another option is the Mandelbrot set, which is the set of parameter \( b \) for which \( f_b^{\circ n}(0)\not\to\infty \), i.e. is bounded.