BACKGROUND: The Mandelbrot set is the set of complex numbers $c$ for which the sequence obtained by iterating $z \mapsto z^2 + c$ starting at $0$ remains bounded. For a complex number $c$, we can define the Julia set $J_c$ similarly, as the set of complex numbers $x$ for which the sequence obtained by iterating $z \mapsto z^2 + c$ starting at $x$ remains bounded. (To see some images of these beautiful sets, look here.)
Anyway, a famous theorem (I think Fatou-Julia) states that $J_c$ is simply connected whenever $c$ is in the Mandelbrot set.
I believe this fact generalizes to multi-brot sets and their corresponding collections of Julia sets, which are described by using a larger integer exponent than 2. For instance, the 3-brot set and corresponding Julia sets $J^3_c$ iterate the cubic $z\mapsto z^3 + c$.
QUESTION: I'm curious about non-integer exponents. I've made some computer drawings of the sets $J^\alpha_c$ for a few non-integer exponents $\alpha$. These technically aren't Julia sets anymore, since the map being iterated is no longer a polynomial, but they still look like gorgeous fractals, although parts of their boundaries appear smooth. But they don't look connected, whether or not $c$ is in the $\alpha$-brot set. Look by the negative real axis (left) in the image below for an example.
I wonder if anyone can confirm my suspicion that the simple connectedness result doesn't generalize to non-integer exponents. If anyone's curious as to the application, the reason simple connectedness is important is because a simply connected set can be drawn extremely efficiently using contour-tracing.
Above: the set $J^\alpha_c$ for $\alpha=3.11$, $c \approx 0.596+0.596i$.