Defining distance in fractal dimensions.

Is it possible define a distance measure in fractal dimensions? namely, what the generalization of

$$D(x,y)=\left(\sum_i(x_i-y_i)^2\right)^{\frac{1}{2}}$$

in fractal dimensions?

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This distance is fine for fractal dimensions (of subsets in $\mathbb R^n$ or even of $l^2$). –  GEdgar Jun 12 '12 at 12:28

When $p \geq 1$ we can use the same formula to define a distance:

$\forall p \in \mathbb R,\, \, p \geq 1$ :

$$d_p(x,y) = \left( \sum_i (x_i-y_i)^p \right)^\frac1p$$

Then we define a distance between points in $\mathbb R^n$.

This is not a distance if $p < 1$.

In fractal geometry and dynamical systems we usually don't use this kind if distances (except for $n=2$). A more interesting distance is the Hausdorff distance between compact sets (could be fractal). Assume $A$ and $B$ two non-empty, compact sets (let's say in $\mathbb R^2$, but works in any metric space), then:

$$d_H(A,B) = \max \left( \inf\{\varepsilon > 0 : A \subset \mathcal V_\epsilon(B) \}, \inf\{\varepsilon > 0 : B \subset \mathcal V_\epsilon(A) \}\right)$$

This is really usefull, for instance we can look on the continuity of Julia sets with respect to some complex parameters, etc.

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