Quasi-symmetric invariant properties of Cantor metric spaces

Abstract

For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasisymmetric invariant properties. The David-Semmes uniformization theorem states that if a compact metric space satisfies all the three properties, then it is quasi-symmetrically equivalent to the middle-third Cantor set. We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic. In this paper, we conclude that for each of exotic types the class of all the conformal gauges of Cantor metric spaces exactly has continuum cardinality. As a byproduct of our study, we state that there exists a Cantor metric space with prescribed Hausdorff dimension and Assouad dimension.

Fractal dimensions and topological embeddings of simplexes into the Gromov--Hausdorff space

“…In [12], the author proved that for all a, b ∈ [0, ∞] with a ≤ b, there exists a Cantor metric space (X, d) with dim H (X, d) = a and dim A (X, d) = b. As a development of this result, we solve the problems of prescribed dimensions for of the five dimensions explained above.…”

“…In this paper, the space (S(m), α ♯ ) is called the (m, α)-Cantor ultrametric space. This space is a generalization of sequentially metrized Cantor spaces defined in the author's paper [12]. Such a construction has been utilized in fractal geometry (for example, [16]).…”

“…We define α ∈ SH by α(n) = 2 −1 . Then, by Lemma 3.6 and Proposition 3.5, and by Theorem 2.8, the space (S(2), α ♯ ) is of the dimension type (1, 1, 1, 1) (see also [12,Lemma 8.8]). Lemma 3.9.…”

“…These constructions have already appeared in [12] as telescope spaces. Note that u[q] and v[q] generate the same topology, which makes Ω be a countable compact metrizable space possessing the unique accumulation point ∞.…”

“…Proof. The construction in this proof has already appeared in [12,Proposition 8.7]. We define α ∈ SH by α(n) = 2 −n 3 .…”