Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 247-266
University of Warsaw, Institute of Mathematics
ul. Banacha 2, 02-097 Warszawa, Poland; baranski 'at' mimuw.edu.pl
Warsaw University of Technology,
Faculty of Mathematics and Information Science
ul. Koszykowa 75, 00-662 Warszawa, Poland;
bkarpin 'at' mini.pw.edu.pl
University of Warsaw, Institute of Mathematics
ul. Banacha 2, 02-097 Warszawa, Poland; A.Zdunik 'at' mimuw.edu.pl
Abstract. In this paper we study relations between the existence of a conformal measure on the Julia set J(f) of a transcendental meromorphic map f and the existence of a zero of the topological pressure function t → P(f,t) for the map f (with respect to the spherical metric). In particular, we show that if f is hyperbolic and admits a t-conformal measure which is not totally supported on the set of escaping points of f, then P(f,t) = 0. On the other hand, for a wide class of maps f, including arbitrary maps with at most finitely many poles and finite set of singular values as well as hyperbolic maps with at most finitely many poles, if P(f,t) = 0, then there exists a t-conformal measure on J(f). This partially answers a question of Mauldin.
2010 Mathematics Subject Classification: Primary 37F10, 37F35; Secondary 28A80.
Key words: Meromorphic functions, Julia sets, conformal measures, topological pressure.
Reference to this article: K. Baranski, B. Karpinska and A. Zdunik: Conformal measures for meromorphic maps. Ann. Acad. Sci. Fenn. Math. 43 (2018), 247-266.
https://doi.org/10.5186/aasfm.2018.4329
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