Ergodicity, numerical range, and fixed points of holomorphic mappings (Q351287)

The authors consider a holomorphic self-map \(F\) on a domain \(D\) and investigate its analytic and geometric structure. They show that, if \(0\) is a quasi-regular fixed point of \(F\) and either \(F\) is power bounded (\(\sup_{x \in B}\|F^n(x)\|\leq M\) for some open ball \(B\) containing \(0\)) or \(F\) is locally uniformly pre-ergodic (\(\lim_{n\to\infty}\|F^n(x)\|/n\to 0\) uniformly on some neighbourhood of \(0\)), then \({\mathcal F}\), the set of fixed points of \(F\), is locally a holomorphic retract. Under certain technical conditions, the set of fixed points of \(F\) is also shown to be a holomorphic retract provided that the real part of the numerical range of \(F\), \(\mathrm{Re}\, V(F)\), is less than or equal to \(1\). In the final section of the paper, a Cartan uniqueness type theorem is obtained, showing that bounds on the numerical range of contractions of a holomorphic function imply that the function is a polynomial of degree at most \(2\).