The standard system of dyadic cubes in the Euclidean space R^n is a collection of half-open cubes of different sizes such that the cubes of every size partition the space and every cube is a finite union of smaller cubes. The construction of this system is very simple but it does rely strongly on the geometrical properties of the space R^n. Hence, if we give up most of the geometrical properties of the space R^n, the construction of sets with similar properties becomes more complicated. In this paper, we show that there exist dyadic cubes in general doubling metric spaces. We do this using certain maximal sets of points and a carefully defined partial order of those points. We look at several different dyadic systems first in R^n and then in general doubling metric spaces.
We start by proving some basic results related to doubling metric spaces and other related topics and continue by introducing the standard, randomized and adjacent systems of dyadic cubes in R^n. Then, in a general doubling metric space, we construct a system of sets that has similar properties as the standard system of dyadic cubes in R^n. We call also these sets cubes although they are not cubes in the usual sense of the word. After this, we add a probabilistic angle to the constructed system by randomizing them and look at two different random systems. Lastly, we look at some applications of the random dyadic systems.
Our goal is to introduce a new simpler way of randomizing dyadic systems in doubling metric spaces and show that this is an effective way of randomizing the systems. We show this by proving that every point in the space has only a small probability of ending up near the boundary of a cube of given size. This property has an interesting application since we can use it to construct systems of Hölder-continuous spline functions in doubling metric spaces. It is still an open problem to prove whether there exist systems of Lipschitz-continuous spline functions in every doubling metric space. We do not know the answer to this problem but we show that at least there exist systems of Hölder-continuous spline functions of every exponent η ∈ (0,1) in every doubling metric space.
