Assume f :U → C is a holomorphic map fixing 0 with derivative λ, where 0 < |λ| 1. If λ is not a root of unity, there is a formal power series φf (z) = z+O(z2) such that φf (λz) = f (φf (z)). This power series is unique and we denote by Rconv( f ) ε [0,+∞] its radius of convergence. We denote by Rgeom( f ) the largest radius r ε [0, Rconv( f )] such that φf (D(0, r )) is a subset of U. In this paper, we present new elementary techniques for studying the maps f → Rconv( f ) and f → Rgeom( f ). Contrary to previous approaches, our techniques do not involve studying the arithmetical properties of rotation numbers.
|Journal||Mathematical Proceedings of the Cambridge Philosophical Society|
|Number of pages||14|
|Publication status||Published - 6 May 2008|
- Siegel disk
- conformal radius