Existence of fixed simple closed curve by polynomials

Question

As the problem mentioned in the title, I wonder that if there exists a simple closed curve on the complex plane which is not circle that can be fixed by a non-linear polynomial with complex coefficients($P(C)=C$, $C$ for the curve and $P$ for the polynomial) ?

I have asked others about this problem, and some said that this is related to dynamical system.

Answer 1

Yes. The Julia set of a polynomial is always fixed in exactly the sense that you say and can often be a fractal, simple, closed curve.

The Julia set is, by definition, the closure of the set of repelling periodic points of the polynomial. For example, if $P(z)=z^2$, then the Julia set of $P$ is exactly the unit circle. If $P(z) = z^2 + c$, however, where $c$ is close to zero, then the Julia set is a somewhat distorted version of the unit circle with a fractal structure. If $c=-1/2$, for example, then the Julia set looks like so:

Thus, that simple closed curve is fixed by $P(z)=z^2-1/2$.

Answer 2

Closed invariant curves :

• orbits inside main chessboard box) in the parabolic case - second image below ( are invariant under proper power of the map) and lay on the simply closed invariant curves
• orbits inside Siegel disc and first image below
• a subset of the Julia set generated by Herman’s Blaschke product has invariant circles - see GEOMETRY OF THE JULIA SET FOR SOME MAPS WITH INVARIANT CIRCLES by Kimberly A. Roth
• ( thx to Claude) on the sphere : 2 external rays landing on the same point and the point at infinity form closed curve, see image by Wolf Jung - third image

All images made with program Mandel by Wolf Jung