If $f:\Omega\rightarrow\mathbb{C}$ is analytic, where $0\in\Omega$, and $f(0) = 0$ and $|f'(0)| = 1$, what other conditions are necessary for finding an invariant open subset of $\Omega$? In particular, I'm wondering about this when $f''(0) = \lambda$ where $|\lambda| < 1$, when can one find a nonempty open subset $U\subset\Omega$ such that $f(U)\subset U$. It isn't necessary to know whether $0\in U$ (in fact, I doubt this is possible in general). I am wondering both
- When does an invariant open set exist?
- Is it possible to construct an invariant open set? (I don't care about a maximal open set here, that I assume is hard. Just any open set.)
I don't know what techniques might be applicable here. I can't even solve it for quadratic $f$. I suppose the full invariant set would be fractal, but I'm just wondering about whether I can find any open set that's invariant. If this is provably not possible to determine based on $f'(0)$ and $f''(0)$, that would be interesting too.