Conjugates of Beta-Numbers and the Zero-Free Domain for a Class of Analytic Functions
Conjugates of Beta-Numbers and the Zero-Free Domain for a Class of Analytic Functions
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DOI：
10.1112/plms/s3-68.3.477
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发表时间：
1994-05
影响因子：
1.8
通讯作者：
B. Solomyak
中科院分区：
文献类型：
--
作者：
B. Solomyak
如果 x = 1 在变换 x ↦ θx (mod 1) 下的轨道是有限的，则实数 θ > 1 是 beta 数。改进 Parry 的结果，我们证明这些数字的所有伽罗瓦共轭都具有小于黄金比例的模数，并且这个估计在模数方面是最好的。结果表明，所有 beta 数的所有共轭集的闭包是闭单位圆盘与函数类的零倒数集的并集。事实证明，这个领域相当奇特；例如，它的边界有一个奇点稠密子集和另一个具有切线的稠密子集。
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.