Conjugates of beta-numbers and the zero-free domain for a class of analytic functions

A real number θ > 1 is a beta-number if the orbit of x = 1 under the transformation x ↦ θx (mod 1) is finite. Refining a result of Parry, we prove that all Galois conjugates of such numbers have modulus less than the golden ratio, and this estimate is best possible in terms of moduli. It is shown that the closure of the set of all conjugates for all beta-numbers is the union of the closed unit disk and the set of reciprocals of zeros of the function class. This domain turns out to be rather peculiar; for instance, its boundary has a dense subset of singularities and another dense subset where it has a tangent.