拟对称映射的最大伸缩商与边界伸缩商

文献1]在研究单位圆周$\pd$上的拟对称自同胚的最大伸缩商与极值最大伸缩商之间的关系时,证明了:如果拟对称自同胚h的最大伸缩商K_q(h)不能在某个以开单位圆△为域、顶点在单位圆周$\pd$上的拓扑四边形Q处达到,则一定有K_q(h)≤H(h)成立,其中H(h)为h的边界伸缩商.这一结论在文献1]中起着重要作用,但证明比较烦琐.该文主要给出该结论一个简单的证明,并且利用这一结论研究拟对称自同胚的最大伸缩商何时可以在某个拓扑四边形上达到.

The Maximal Dilatation and Boundary Dilatation of Quasi-symmetric Mapping

To study the relationship between the maximal dilatation of quasi-symmetric self-homoemorphism of the unit circle and the maximal dilatation of its extremal extension, it was proved in 1] that if the maximal dilatation K_q(h) of a qusi-symmetric self-homoemorphism h cannot be arrive at some quadrilateral with the unit disk △ as its domain and vertices on the unit circle $\pd$, then K_q(h)≤ H(h), where H(h) is the boundary dilatation of h. The main resluts in 1] quite depend on this result. But its proof there is very complicated. In this paper, the authors give another elementary and simple proof. Furthermore, they use this result to study the problem when the maximal dilatation of a qusi-symmetric self-homoemorphism of the unit circle could be arrive at some quadrilateral.