For connectivity of random geometric graphs, where there is no density for underlying distribution of the vertices, we consider n i.i.d. Cantor distributed points on [0, 1]. We show that for this random geometric graph, the connectivity threshold Rn, converges almost surely to a constant 1−2φ where 0 < φ < 1/2, which for standard Cantor distribution is 1/3. We also show that ‖Rn − (1− 2φ)‖1 ∼ 2...

We show that every minimal, free action of the group Z2 on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, Z-actions and Z2-actions.

A nilpotent Cantor action is a minimal equicontinuous $\Phi \colon \Gamma \times \mathfrak {X} \to {X}$ on space $\mathfrak {X}$, where $\Gamma$ contains finitely-generated subgroup $\Gamma _0 \subset \Gamma$ of finite index. In this note, we show that these actions are distinguished among general actions: any effective finitely generated group space, which continuously orbit equivalent to acti...

We survey the some of the main results, ideas and conjectures concerning two problems and their connections. The first problem concerns determining when two Bernoulli trial measures are homeomorphic to each other, i.e. when one is the image measure of the other via a homeomorphism of the Cantor space. The second problem concerns the following. Given a positive integer k characterize those Berno...

A generating IFS of a Cantor set F is an IFS whose attractor is F . For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in R under t...

In analogy to ordinary q-additive functions based on q-adic expansions one may use Cantor expansions with a Cantor base Q to define (strongly) Q-additive functions. This paper deals with distribution properties of multi-dimensional sequences which are generated by such Q-additive functions. If in each component we have the same Cantor base Q, then we show that uniform distribution already impli...

Although every Cantor subset of the circle (S) is the minimal set of some homeomorphism of S, not every such set is minimal for a C diffeomorphism of S. In this work, we construct new examples of Cantor sets in S that are not minimal for any C-diffeomorphim of S.