Fix-points of meromorphic functions and iterated entire functions

Abstract

Let f be a rational function of degree k and let g be a transcendental meromorphic function. Gross and Osgood [13] proved that the composite function fog has infinitely many fix-points if k > 3. If k = 2 then fog may have only a finite number of fix-points and all such f and g have been characterized by Gross and Osgood. There is a corresponding result for the case that f is a transcendental meromorphic function and that g is a polynomial. In fact the two cases are closely connected since, as pointed out by Gross and Yang [ 14, p. 214, Proof of Theorem 23, fog has infinitely many fix-points if and only if gof does. This paper deals with the fix-points offog, wherefand g are both transcendental. It has been proved in [6] that f 0 g has infinitely many fix-points if f and g are entire. This confirmed a conjecture of Gross [ 111. There are a number of papers [S, 12, 14, 22, 27, 283 besides [6] dealing with fix-points of fog for entire f and g, but their methods do not seem to extend to the case of meromorphic f and in fact very little seems to be known about this case. Gross [ 1 l] has asked whether fog has infinitely many fix-points if f is transcendental and meromorphic and if g is transcendental and entire. Using some ideas from [6] we shall prove that this is indeed the case iffsatisfies a certain condition to be discussed afterward.