I used to write that when a set has a finite fractal dimension then it could be treated as a finite dimesional set. But, in fact, I don’t really understand what it mean (lol).
Recently, I’ve read in a paper (Large time behavior via method of l-trajectory), and have known the important result:
If is a compact metric space such that , then there exists an injective Lipschitz continuous mapping such that its inverse is Holder continuous. In other words, if , then is placed in the graph of a Holder continuous mapping that maps the compact subset of onto . Moreover, if is a subset of a Hilbert space then can in addition be an orthogonal projector.
(The proof can be seen in a work of Foias and Olson)