linguistic:translation, information:filtering
In general metric spaces

In a general metric space (X, d), an exact similitude is a function f from the metric space X into itself that multiplies all distances by the same positive scalar r, called f 's contraction factor, so that for any two points x and y we have

Weaker versions of similarity would for instance have f be a bi- Lipschitz function and the scalar r a limit

This weaker version applies when the metric is an effective resistance on a topologically self-similar set.

A self-similar subset of a metric space (X, d) is a set K for which there exists a finite set of similitudes { f } with contraction factors 0 ≤ r < 1 such that K is the unique compact subset of X for which

These self-similar sets have a self-similar measure μ with dimension D given by the formula

which is often (but not always) equal to the set's Hausdorff dimension and packing dimension. If the overlaps between the f (K) are "small", we have the following simple formula for the measure: