Blog

# 2017-4 Blog Entry List

## トポロジーセミナー（2017/04/27）

アブストラクト：For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties.
We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic.
For instance, the middle-third Cantor set is standard.
In this talk, we discuss our constructions of exotic Cantor metric spaces for all the possible cases of satisfying each of the three properties or not.
Our constructions enable us to classify Cantor metric spaces into eight types with concrete examples.
The David-Semmes uniformization theorem tells us that standard Cantor metric spaces are quasi-symmetric equivalent.
In this talk, we conclude that there exist at least two exotic Cantor metric spaces of the same type that are not quasi-symmetric equivalent to each other.
Moreover, for each of all the non-uniformly disconnected types, there exist at least aleph one many quasi-symmetric equivalent classes of Cantor metric spaces of such a given type.
As a byproduct of our study, we state that there exists a Cantor metric space with prescribed Hausdorff dimension and Assoud dimension.