My textbook tells me that for a polynomial $P$, its Julia set $J(P)$ is the nuit circle iff $P(z)=az^n$,where $\lvert a\rvert =1$,and $n\geq 2$.
So I want to know whether there is a Julia set of a polynomial is the boundary of a square. Or more generally, whether there is such a rational map.
I found a paper stating that under certain conditions, the Julia set of rational mappings is a Jordan curve. But for a specific square, whether or not there is a corresponding rational map, that's my puzzle.
For a polynomial, the answer is no. By a theorem of Zdunik, any connected Julia set either is a fractal (in the sense that its Hausdorff dimension is strictly larger than one) or it's a segment or the unit circle (and in both cases, we know what polynomials those are).
I am fairly sure the unit square is not the Julia set of any rational map but I don't have any argument in mind.