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?
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?
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.