Parabolic meromorphic functions

Zheng Jian-Hua

Vol. 250 (2011), No. 2, 487–509

DOI: 10.2140/pjm.2011.250.487

Abstract

Following the definition of parabolic rational functions and in view of the behavior of transcendental meromorphic functions, we give the definition of parabolic transcendental meromorphic functions. We discuss their dynamical behavior and prove the existence of conformal measures and invariant measures over their Julia sets, thus extending Denker and Urbański’s work on parabolic rational functions. However, our method for proving the existence of the conformal measures differs in that we use the Perron–Frobenius–Ruelle operator.

Keywords

parabolic meromorphic functions, expansive map, conformal measures, pressure functions, Poincaré exponents

Mathematical Subject Classification 2000

Primary: 30B20, 30D05, 30D10

Secondary: 34M05, 37F10

Milestones

Received: 11 January 2010

Revised: 5 November 2010

Accepted: 10 November 2010

Published: 31 March 2011