Blaschke products and expanding maps of the circle
Blaschke products and expanding maps of the circle
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Blaschke 产品和扩展的圈子地图
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发表时间：
1999
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通讯作者：
D. Tischler
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作者：
D. Tischler
对于闭单位圆盘到自身上的解析映射，给出了边界圆不变的边界映射扩张的充要条件。设B是从闭单位圆盘到其自身的解析映射，它将边界圆映射到其自身。假设B在限制于边界时膨胀。在这种情况下，已知在圆盘的内部存在一个固定点。然而，匡威就不正确了。本文给出了B在边界圆上扩张的几个充要条件，其中一个条件给出了扩张映射的导数的估计。上面的映射B是有限Blaschke积的常数倍。设Dr = {z ∈ C：|z| 1，z ∈ C1。ii）对于z ∈ C1，对某个r1 1。ii）设A = {r：Ba（Dr）包含在Dr的内部}。则A是以1为端点的区间。定理1的证明。（i）第（ii）段。对z ∈ C1，有B′ a（z）= τ ′ a（z）.因此，Ba垂直于C1的导数大于1。由于C1是Ba不变量，当r足够接近1时，Ba（Dr）是Dr. ii）的内部。这是由布劳威尔不动点定理和λ · Ba满足ii）的事实得出的。（iii）第一节）。Ba在黎曼球面上有n+1个不动点。注意，对于所有z ∈ C，B（1 z）B（z）= 1。如果Ba在D1的内部有一个不动点，那么它在1998年4月14日编辑接收中有一个。1991年数学学科分类。初级58 F03，30 D50。1999年美国数学学会American Mathematical Society
For an analytic map of the closed unit disk onto itself that leaves the boundary circle invariant, we give necessary and sufficient conditions for the map of the boundary to be expanding. Let B be an analytic map from the closed unit disk onto itself that maps the boundary circle into itself. Suppose B is expanding when restricted to the boundary. In this case, it is known that there is a fixed point in the interior of the disk. However the converse is not true. In this note we will give a couple of necessary and sufficient conditions for B to be expanding on the boundary circle, one of which gives an estimate on the derivative of the expanding map. A map B as above is a constant multiple of a finite Blaschke product. Let Dr = {z ∈ C : |z| 1 for z ∈ C1. ii) For some r1 1 for z ∈ C1. ii) Let A = {r : Ba(Dr) is contained in the interior of Dr}. Then A is an interval with 1 as an endpoint. Proof of Theorem 1. i) ⇒ ii). For z ∈ C1, B′ a(z) = τ ′ a(z). Therefore, the derivative of Ba normal to C1 is greater than 1. Since C1 is invariant by Ba, for r sufficiently close to 1, Ba(Dr) ⊂ interior of Dr. ii) ⇒ iii). This follows from the Brouwer fixed point theorem and the fact that λ · Ba satisfies ii) whenever Ba does. iii)⇒i). Ba has n+1 fixed points on the Riemann sphere. Note that B(1z )B(z) = 1 for all z ∈ C. If Ba has a fixed point in the interior of D1, then it has one in Received by the editors April 14, 1998. 1991 Mathematics Subject Classification. Primary 58F03, 30D50. c ©1999 American Mathematical Society