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Results tagged with dimension-theory
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user 82839
Hausdorff dimension, box dimension, packing dimension and similar concepts.
0
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Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ [closed]
Let $f:[0,1] \to \mathbb{R}, G = graph(f)$.
If $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ for all partitions $0 = x_0< \ldots < x_m = 1 $ then $H^s(G) < \infty$
What technique can I use to prove t …
CommunityBot
- 1
7
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2
answers
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Box dimension of the graph of an increasing function
This Hausdorff dimension of the graph of an increasing function shows that:
Let $f$ be a continuous, strictly increasing function from $[0,1]$ to
itself with $f(0)=0, f(1)=1$. Then $dim_H \; G …
- 439
3
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Box dimension of the graph of an increasing function
Let me encode the solution of the problem explicitly. From the linked answer we had:
Theorem If $\gamma:[a,b]\to (X,d)$ is an injective rectifiable curve and $\Gamma=f([a,b])$, then $$\mathcal{H} …
- 439