Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-28T21:42:23.309Z Has data issue: false hasContentIssue false

No elliptic limits for quadratic maps

Published online by Cambridge University Press:  01 February 1999

CARSTEN LUNDE PETERSEN
Affiliation:
IMFUFA, Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark (e-mail: lunde@.ruc.dk)

Abstract

We establish bounds for the multipliers of those periodic orbits of $R_\mu(z) = z(z+\mu)/(1+\overline\mu z) $, which have a Poincaré rotation number $ p/q $. The bounds are given in terms of $ p/q $ and the (logarithmic) hororadius of $\mu$ to $e^{2\pi ip/q} $. The principal tool is a new construction denoted a ‘star’ of an immediate attracting basin. The bounds are used to prove properties of the space of Möbius conjugacy classes of quadratic rational maps. These properties are related to the mating and non-mating conjecture for quadratic polynomials lsqb;Ta]. Moreover they are also reminiscent of Chuckrows theorem on the non-existence of elliptic limits of loxodromic elements in quasiconformal deformations of Kleinian groups. We bear this analogy further by proving an analog of Chuckrows theorem for deformations of certain holomorphic maps.

Type
Research Article
Copyright
1999 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)