Similarity in general metric spaces Similarity (geometry)

sierpiński triangle. space having self-similarity dimension log 3/log 2 = log23, approximately 1.58. (from hausdorff dimension.)

in general metric space (x, d), exact similitude function f metric space x multiplies distances same positive scalar r, called f s contraction factor, 2 points x , y have

d
(
f
(
x
)
,
f
(
y
)
)
=
r
d
(
x
,
y
)
.




{\displaystyle d(f(x),f(y))=rd(x,y).\,\,}

weaker versions of similarity instance have f bi-lipschitz function , scalar r limit

lim



d
(
f
(
x
)
,
f
(
y
)
)


d
(
x
,
y
)



=
r
.


{\displaystyle \lim {\frac {d(f(x),f(y))}{d(x,y)}}=r.}

this weaker version applies when metric effective resistance on topologically self-similar set.

a self-similar subset of metric space (x, d) set k there exists finite set of similitudes { fs }s∈s contraction factors 0 ≤ rs < 1 such k unique compact subset of x which

⋃

s
∈
s



f

s


(
k
)
=
k
.



{\displaystyle \bigcup _{s\in s}f_{s}(k)=k.\,}

these self-similar sets have self-similar measure μ dimension d given formula

∑

s
∈
s


(

r

s



)

d


=
1



{\displaystyle \sum _{s\in s}(r_{s})^{d}=1\,}

which (but not always) equal set s hausdorff dimension , packing dimension. if overlaps between fs(k) small , have following simple formula measure:

μ

d


(

f


s

1




∘

f


s

2




∘
⋯
∘

f


s

n




(
k
)
)
=
(

r


s

1




⋅

r


s

2




⋯

r


s

n





)

d


.



{\displaystyle \mu ^{d}(f_{s_{1}}\circ f_{s_{2}}\circ \cdots \circ f_{s_{n}}(k))=(r_{s_{1}}\cdot r_{s_{2}}\cdots r_{s_{n}})^{d}.\,}