CONJUGATES OF BETA-NUMBERS AND THE ZERO-FREE DOMAIN FOR A CLASS OF ANALYTIC-FUNCTIONS

A real number theta > 1 is a beta-number if the orbit of x = 1 under t he transformation x bar arrow pointing right thetax (mod 1) is finite. Refining a result of Parry, we prove that all Galois conjugates of su ch 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 t he set of all conjugates for all beta-numbers is the union of the clos ed unit disk and the set of reciprocals of zeros of the function class {f(z) = 1 + SIGMA a(j)z(j), 0 less-than-or-equal-to a(j) less-than-or -equal-to 1}. This domain turns out to be rather peculiar; for instanc e, its boundary has a dense subset of singularities and another dense subset where it has a tangent.