Geometry and dynamics of some meromorphic functions

✍ Scribed by Janina Kotus; Mariusz Urbański

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 255 KB
• Volume: 279
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410437

✦ Synopsis

Abstract

Themeromorphic maps f~λ~ (z) = λ (1 – exp(–2__z__))^–1^, λ > 0, of the complex plane are thoroughly investigated. With each map f~λ~ associated is its projection F~λ~ on the infinite cylinder Q. This map and the set J~r~ (F~λ~) consisting of those points in the cylinder Q whose ω ‐limit set under F~λ~ is not contained in the set {0, –∞} will form the primary objects of our interest in this article. Let h~λ~ = HD(J~r~ (F~λ~)) be the Hausdorff dimension of J~r~ (F~λ~). We prove that h~λ~ ∈ (1, 2). The h~λ~ ‐dimensional Hausdorff measure H__h~λ~__ of J~r~ (F~λ~) is proven to be positive and finite. The h~λ~ ‐dimensional packing measure of J~r~ (F~λ~) is shown to be locally infinite at every point of this set. There exists a unique Borel probability F~λ~ ‐invariant measure μ~λ~ on J~r~ (Fλ) absolutely continuous with respect to the Hausdorff measure H__h~λ~. This measure turns out to be ergodic and equivalent to H__h~λ~. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)