Self-similar conductance patterns in graphene Cantor-like structures

Self-Similar Markov Processes on Cantor Set

We define analogues of Brownian motion on the triadic Cantor set by introducing a few natural requirements on the Markov semigroup. We give a detailed description of these symmetric self-similar processes and study their properties such as mixing and moment asymptotics.

Self-similar Measures and Intersections of Cantor Sets

It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-α Cantor set and α ∈ ( 1 3 , 1 2 ). We show that for any compact set K and for a.e. α ∈ (0, 1), the arithmetic sum of K and the middle-α Cantor set does indeed have positive...

Correlation dimension for self-similar Cantor sets with overlaps

We consider self-similar Cantor sets Λ ⊂ R which are either homogeneous and Λ− Λ is an interval, or not homogeneous but having thickness greater than one. We have a natural labeling of the points of Λ which comes from its construction. In case of overlaps among the cylinders of Λ, there are some “bad” pairs (τ, ω) of labels such that τ and ω label the same point of Λ. We express how much the co...

The Cherenkov Effect in Graphene-Like Structures

Dilatation Structures Ii. Linearity, Self-similarity and the Cantor Set