We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rank-one symmetric spaces of non-compact type among locally compact and connected metric spaces. Furthermore, we investigate the metric implications of homogeneity with respect to uniformly strongly quasi-Möbius self-homeomorphisms, connecting such homogeneity with the combination of uniform bi-Lipschitz homogeneity and quasi-invertibility. In this context we characterize spaces containing a cut point and provide several metric properties of spaces containing no cut points. These results are motivated by a desire to characterize the snowflakes of boundaries of rank-one symmetric spaces up to bi-Lipschitz equivalence.

中文翻译：

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求可逆齐次度量空间的拟Möbius刻画

我们研究关于Möbius自同胚性享有局部同质性的局部紧凑度量空间。我们研究这种同质性与等距同质性与可逆性的组合之间的联系。特别是，我们提供了局部紧凑和连通度量空间中非紧凑型秩一对称空间边界的雪花的新表征。此外，我们研究了均匀性关于均匀强拟莫比乌斯自同胚性的度量含义，并将这种均匀性与均匀双Lipschitz均匀性和准可逆性的组合联系起来。在这种情况下，我们表征包含切点的空间，并提供不包含切点的空间的若干度量属性。