Dynamics of slowly growing entire functions

Abstract

Dedicated to George Szekeres on his 90th birthdayFor a transcendental entire function f let M(r) denote the maximum modulus of f(z) for |z| = r. Then A(r) = log M(r)/logr tends to infinity with r. Many properties of transcendental entire functions with sufficiently small A(r) resemble those of polynomials. However the dynamical properties of iterates of such functions may be very different. For instance in the stable set F(f) where the iterates of f form a normal family the components are preperiodic under f in the case of a polynomial; but there are transcendental functions with arbitrarily small A(r) such that F(f) has nonpreperiodic components, so called wandering components, which are bounded rings in which the iterates tend to infinity. One might ask if all small functions are like this.A striking recent result of Bergweiler and Eremenko shows that there are arbitrarily small transcendental entire functions with empty stable set—a thing impossible for polynomials. By extending the technique of Bergweiler and Eremenko, an arbitrarily small transcendental entire function is constructed such that F is nonempty, every component G of F is bounded, simply-connected and the iterates tend to zero in G. Zero belongs to an invariant component of F, so there are no wandering components. The Julia set which is the complement of F is connected and contains a dense subset of “buried’ points which belong to the boundary of no component of F. This bevaviour is impossible for a polynomial.