## The importance of fractal dimension

I used to write that when a set $A$ 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 $C$ is a compact metric space such that $d_f^C(C) , then there exists an injective Lipschitz continuous mapping $P: C\mapsto \mathbb R^m$ such that its inverse is Holder continuous. In other words, if $d_f^C(C) , then $C$ is placed in the graph of a Holder continuous mapping that maps the compact subset of $\mathbb R^m$ onto $C$. Moreover, if $C$ is a subset of a Hilbert space then $P$ can in addition be an orthogonal projector.

(The proof can be seen in a work of Foias and Olson)