In the same way that we can say manifolds are made of pieces that look like $\mathbb{R}^n$, is there any way to say that spaces with the same hausdorff dimension are made up of pieces that look the same? Formally, I mean something like:
For $\alpha \in \mathbb{R}^+$, consider the category of (compact, ...) spaces with Hausdorff dimension $\alpha$ where the maps are bi-lipshitz. Is there an object in this category with a bi-lipshitz map to every other?
The reason for the maps being bi-lipshitz is that if we have a bi-lipshitz map $f : X \to Y$, then the hausdorff dimension of the domain is the same as its image $\text{dim}_H (X) = \text{dim}_H (f(X))$. In particular, this statement is saying that for every $\alpha$ there is a 'prototypical' space of dimension $\alpha$ such that all spaces with dimension $\alpha$ have a piece that looks like the prototype.
A (further) condition that I'm not sure is necessary but might help is to only consider spaces which are 'everywhere dimension $\alpha$', in that all balls in the space have dimension $\alpha$. Even in the $\alpha = 1$ case, (where you would hope that if the spaces are all compact and everywhere dimension $1$ that the initial object is a real interval?) I'm finding it hard to prove/find counterexample
