论文标题
无界域上非专用映射的通用属性
Generic properties of nonexpansive mappings on unbounded domains
论文作者
论文摘要
我们研究了无界的完整双曲线度空间上非专用映射的典型特性。对于两个有界集合的统一收敛指标家族，我们表明，典型的非专用映射是每个有界子集的rakotch收缩，并且有一个有限的集合，该集合被此映射映射到自身。特别是，我们获得此设置中典型的非统计映射具有唯一的固定点，可以通过迭代映射来达到该点。然而，事实证明，典型的映射不是整个空间上的rakotch收缩，并且在其域的残留子集上具有最大的Lipschitz常数。通常，我们的意思是，与此属性的一组映射的补充是$σ$ - $ ϕ $ - 孔子，也就是说，从公制意义上讲很小。对于指标的融合度量，我们表明一组Rakotch收缩是微不足道的。
We investigate typical properties of nonexpansive mappings on unbounded complete hyperbolic metric spaces. For two families of metrics of uniform convergence on bounded sets, we show that the typical nonexpansive mapping is a Rakotch contraction on every bounded subset and that there is a bounded set which is mapped into itself by this mapping. In particular, we obtain that the typical nonexpansive mapping in this setting has a unique fixed point which can be reached by iterating the mapping. Nevertheless, it turns out that the typical mapping is not a Rakotch contraction on the whole space and that it has the maximal possible Lipschitz constant of one on a residual subset of its domain. By typical we mean that the complement of the set of mappings with this property is $σ$-$ϕ$-porous, that is, small in a metric sense. For a metric of pointwise convergence, we show that the set of Rakotch contractions is meagre.
