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) <m/2, m\in \mathbb N, 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) <m/2, m\in \mathbb N, 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)

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About baotangquoc

Lecturer School of Applied Mathematics and Informatics Hanoi University of Science and Technology No 1, Dai Co Viet Street, Hanoi
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