Related papers: Characterizing the Cantor bi-cube in asymptotic ca…

By introducing new deformations on symbolic Cantor sets and ultrametric spaces, we prove that doubling ultrametric spaces admit bilipschitz embedding into Cantor sets. If in addition the spaces are uniformly perfect, we show that they are…

We prove that each coarsely homogenous separable metric space $X$ is coarsely equivalent to one of the spaces: the sigleton, the Cantor macro-cube or the Baire macro-space. This classification is derived from coarse characterizations of the…

For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David-Semmes uniformization theorem states that if a compact metric space satisfies…

The purpose of this note is twofold. In the first part we observe that two finitely generated non-amenable groups are quasi-isometric if and only if they admit topologically orbit equivalent Cantor minimal actions. In particular, free…

We prove that two homogeneous ultra-metric spaces $X,Y$ are coarsely equivalent if and only if $\mathrm{Ent}^\sharp(X)=\mathrm{Ent}^\sharp(Y)$ where $\mathrm{Ent}^\sharp(X)$ is the so-called sharp entropy of $X$. This classification implies…

This paper concerns the self-similarity of topological spaces, in the sense defined in math.DS/0411344. I show how to recognize self-similar spaces, or more precisely, universal solutions of self-similarity systems. Examples include the…

Inspired by a classical theorem of topological dimension theory, we prove that every geodesic metric space of asymptotic dimension $n$ containing a bi-infinite geodesic can be coarsely separated by a subset $S$ of asymptotic dimension equal…

Prompted by a recent question of G. Hjorth as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we…

In spaces of metrics, we investigate topological distributions of the doubling property, the uniform disconnectedness, and the uniform perfectness, which are the quasi-symmetrically invariant properties appearing in the David--Semmes…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

Let mu(r) be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights r and 1-r. It is a long-standing open problem to characterize those r and s such that mu(r) and mu(s) are topologically…

We characterize $n$-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite $n$-densities and one of the following: is an $n$-dimensional Lipschitz…

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an…

We show that two cookie-cutter Cantor sets with the same symbolic coding are differentiably equivalent if and only if their Hausdorff dimensions are equal.

We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincar\'{e} inequality. The two main results we obtain are a decomposition theorem…

We introduce the notion of tiling spaces for metric spaces. The class of tiling spaces contains the Euclidean spaces, the middle-third Cantor set, and various self-similar spaces appearing in fractal geometry. For doubling tiling spaces, we…

We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article we give a complete characterization of complete metric…

By using a similar pattern of arguments, we show that in three categories the collection of isomorphisms forms a residual subset of the space of morphisms. We first consider surjective continuous mappings on Cantor spaces. Next, we look at…

Compact metric spaces form an important class of metric spaces, but the category that they define lacks many important properties such as completeness and cocompleteness. In recent studies of "metric domain theory" and Stone-type dualities,…

A class of ultrametric Cantor sets $(C, d_{u})$ introduced recently in literature (Raut, S and Datta, D P (2009), Fractals, 17, 45-52) is shown to enjoy some novel properties. The ultrametric $d_{u}$ is defined using the concept of {\em…