度量空间中拟双曲映射的局部性质

Local properties of quasi-hyperbolic mapping in metric spaces

为了研究度量空间中J.Väisälä提出的拟双曲映射是否具有从局部到整体的性质,首先,利用拟双曲映射的性质,证明拟双曲映射的逆映射是一个完全拟双曲映射;其次,为了保证曲线分点的个数为有限数,引入了拟John-ball域的概念,从而建立局部拟双曲度量与整体拟双曲度量之间的关系;最后,在两个适当的拟凸度量空间之间,证明半局部的拟双曲映射是全局的拟双曲映射。

In order to study the problem proposed by J.Väisälä,that is,whether quasi-hyperbolic mapping has properties from local to global in the metric space.First of all,we proved that the inverse mapping of the quasi-hyperbolic mapping is a fully quasi-hyperbolic mapping by using the properties of the quasi-hyperbolic mapping.Secondly,the concept of quasi John-ball domain was introduced to ensure that the number of curve points was a finite number.Thereby,we established the relationship between the local quasi-hyperbolic metric and the global quasi-hyperbolic metric.Finally,we proved that the semi-local quasi-hyperbolic mapping is a global quasi-hyperbolic maping between two appropriate quasi-convex metric spaces.