For metric spaces, the doubling property, the uniform disconnectedness, andthe uniform perfectness are known as quasi-symmetric invariant properties. TheDavid-Semmes uniformization theorem states that if a compact metric spacesatisfies all the three properties, then it is quasi-symmetrically equivalentto the middle-third Cantor set. We say that a Cantor metric space is standardif it satisfies all the three properties; otherwise, it is exotic. In thispaper, we conclude that for each of exotic types the class of all the conformalgauges of Cantor metric spaces has continuum cardinality. As a byproduct of ourstudy, we state that there exists a Cantor metric space with prescribedHausdorff dimension and Assouad dimension.
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