Embedding of the snowflake metric spaces into the euclidean spaces as self-similar sets

Let (X, d) be a metric space and (X, d^α) be the so-called snowflaked version of (X, d) whereby 0 < α < 1 and d^α(x, y) = d(x, y)^α for all x, y ∈ X. On the other hand, whether a metric space can be embedded bi-Lipschitzly in an Euclidean space is a very difficult and important problem. One of the beautiful theorems on metric embeddings is the Assouad Theorem. In his 1983 paper, Assouad proved that (X, d) is a doubling metric space if and only if the snowflake metric space (X, d^α) can be embedded bi-Lipschitzly into an Euclidean space. In that study, the author describe three different methods to obtain an embedding, one of these methods is given by the notion of so-called generalized Koch chains. An important instance of this theorem is the case (X = [0, 1], d) where d is the standard metric for which Assouad gives a special proof and determines the optimal dimension of the ambient Euclidean space. In this thesis, we give another alternative proof, which uses the notion of iterated function system, in the case X = [0, 1] equipped with the standard metric. In Section 1, we introduce the snowflake metric spaces and Assoud Theorem and in Section 2 we introduce the notions of iterated function system and fractal dimension. And in the last section, we show that ([0, 1], d^α), can be embedded bi-Lipschitzly into the [|1/α|]+ 1-dimensional Euclidean space as a self-similar set which is an attractor of an iterated function system whose fractal dimension is 1 α for each α. Keywords: Assouad's Theorem, Bi-Lipschitz embedding, Snowflake metric spaces, Self-similar set, Iterated function systems