Monatsh. Math. 149, 103–117 (2006) DOI 10.1007/s00605-005-0373-5 Elliptic Functions With Critical Points Eventually Mapped Onto Infinity By Janina Kotus Warsaw University of Technology, Poland Communicated by K. Schmidt Received April 6, 2005; accepted in revised form August 16, 2005 Published online February 27, 2006 # Springer-Verlag 2006 Abstract. We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto 1. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f . We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) -finite f -invariant measure  equivalent to m. The measure  is ergodic and conservative. 2000 Mathematics Subject Classification: 37F35, 37F10, 30D05 Key words: Holomorphic dynamics, elliptic functions, Julia sets, Hausdorff dimension, conformal measure, ergodicity 1. Introduction Throughout the entire paper let f : C ! C denote a non-constant elliptic func- tion. Every such function is doubly periodic and meromorphic. In particular there exist two vectors w 1 ; w 2 , Im  w 1 w 2  6¼ 0, such that for every z 2 C and m; n 2 Z, f ðzÞ¼ f ðz þ mw 1 þ nw 2 Þ. Let R ¼ft 1 w 1 þ t 2 w 2 : 0 4 t 1 ; t 2 4 1g; be the basic fundamental parallelogram of f . The Fatou set Fðf Þ of a meromorphic function f : C ! C is defined in exactly the same manner as for rational functions; Fðf Þ is the set of points z 2 C such that all the iterates are defined and form a normal family on a neighborhood of z. The Julia set J ðf Þ is the complement of Fðf Þ in C. In particular, since f has infinitely many poles, J ð f Þ¼ [ n 5 0 f n ð1Þ: The research was supported in part by the Polish KBN Grant no. 2 PO3A 034 25, the Warsaw University of Technology Grant no. 504G 11200043000 and by the NSF=PAN grant INT-0306004.