Acta Math. Hungar. 73 (3) (1996),213-218. ERGODIC INVARIANT PROBABILITY MEASURES AND ENTIRE FUNCTIONS J. P. R. CHRISTENSEN (Copenhagen) and P. FISCHER (Guelph) 1. Introduction The concept of invariant measures plays a fundamental role in ergodic theory and in the theory of dynamical systems. In this work a family of ergodic invariant non-discrete probability measures will be constructed for non-linear entire functions. To a given relatively open non-empty subset U of the Julia set of an entire function f an invariant probability will be constructed whose support intersects U in a non-empty set. The following notation will be used. If f is a function then fol = f, fo2 = f o f,..., fo(=+l) = f o fon,.., for n = 1, 2,... denote its successive iterates and fo0 stands for the identity mapping. Let ~={(xl,x2,...,xn,...):zi=O or xi=l forall iEN}, and let S be the shift (operator) on ~. The topology on ~ is the usual product topology (of discrete topologies). The set of all (Borel) measurable subsets of ~ will be denoted by B. The Julia set of an entire function f is denoted by if(f). The closure of a set A is designated by A, and its boundary by cOA. For a complex number a and a positive number r the open disc with center a and radius r { z E C: [z -a[ < r} is denoted by D(a;r), and n stands for D(0; 1). The set of all analytic functions in the unit disc will be denoted by H(D). We shall say that f is a non-linear entire function when it is an entire function which is not a polynomial of degree less than or equal to one. A generalized version of Bloch's lemma plays an important role in this paper. This version is due to Ahffors. It is stated here only in the special case in which it will be used. We are following closely the formulation of I. N. Baker [2]. LEMMA 1. Let w = f(z) be regular in the disc Izl < R and let El, E2 and E3 be circular discs of the w plane such that the closures of the Ei are mutually disjoint. Then there is a constant Co which depends only on the 0236-5294/96/$5.00 ~) 1996 Akad~mial Kiad6, Budapest