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Dimensiune Hausdorff .
În cadrul topologiei , dimensiunea Hausdorff este un număr real pozitiv, asociat unui spațiu metric și extinde noțiunea de dimensiune a unui spațiu vectorial real. A fost introdusă în 1918 de către Felix Hausdorff și dezvoltată ulterior de către Abram Samoilovici Bezicovici, de unde și denumirea de dimensiune Hausdorff-Bezicovici .
Triunghiul lui Sierpinski, un spaţiu având dimensiunea fractală ln 3/ln 2, ori log2 3, care este circa 1,58. Dimensiunea Hausdorff ne oferă un mijloc uzual de calculare a dimensiunii unui spațiu metric .
H
δ
s
(
E
)
=
inf
{
∑
i
=
1
∞
d
i
a
m
(
A
i
)
s
}
{\displaystyle H_{\delta }^{s}(E)=\inf \left\{\sum _{i=1}^{\infty }diam(A_{i})^{s}\right\))
H
s
(
E
)
=
lim
δ
→
0
H
δ
s
(
E
)
{\displaystyle H^{s}(E)=\lim _{\delta \rightarrow 0}H_{\delta }^{s}(E)}
dim
H
(
E
)
=
inf
{
s
,
H
s
(
E
)
=
0
}
=
sup
{
s
,
H
s
(
E
)
=
∞
}
{\displaystyle \dim _{H}(E)=\inf \left\{s,H^{s}(E)=0\right\}=\sup \left\{s,H^{s}(E)=\infty \right\))
Determinarea dimensiunii Hausdorff pentru intervalul
X
=
[
0
,
1
]
⊂
R
{\displaystyle X=[0,1]\subset \mathbb {R} }
:
Pentru
s
>
1
{\displaystyle s>1\,}
Pentru
ε
>
0
{\displaystyle \varepsilon >0\,}
, fie numărul natural
N
ε
{\displaystyle N_{\varepsilon ))
astfel ales încât
1
N
ε
<
ε
{\displaystyle {\frac {1}{N_{\varepsilon ))}<\varepsilon }
.
Cu acoperirea specială
A
i
=
[
i
−
1
N
ε
,
i
N
ε
]
{\displaystyle A_{i}={\big [}{\frac {i-1}{N_{\varepsilon ))},{\frac {i}{N_{\varepsilon ))}{\big ]))
pentru
1
≤
i
≤
N
ϵ
,
A
i
=
1
{\displaystyle 1\leq i\leq N_{\epsilon },A_{i}=1}
pentru
i
≥
N
ε
{\displaystyle i\geq N_{\varepsilon ))
.
Urmează
H
ε
s
(
X
)
≤
N
ε
⋅
(
1
N
ε
)
s
=
(
1
N
ε
)
s
−
1
<
ε
s
−
1
{\displaystyle H_{\varepsilon }^{s}(X)\leq N_{\varepsilon }\cdot {\big (}{\frac {1}{N_{\varepsilon ))}{\big )}^{s}={\big (}{\frac {1}{N_{\varepsilon ))}{\big )}^{s-1}<\varepsilon ^{s-1))
.
Pentru
s
<
1
{\displaystyle s<1\,}
Deoarece
d
(
A
i
)
<
ε
{\displaystyle d(A_{i})<\varepsilon }
, avem:
∑
d
(
A
i
)
s
=
∑
d
(
A
i
)
d
(
A
i
)
1
−
s
>
∑
d
(
A
i
)
ε
1
−
s
{\displaystyle \sum d(A_{i})^{s}=\sum {\frac {d(A_{i})}{d(A_{i})^{1-s))}>\sum {\frac {d(A_{i})}{\varepsilon ^{1-s))))
.Cum însă
A
i
{\displaystyle A_{i}\,}
intervalul
X
{\displaystyle X\,}
acoperă, suma tuturor diametrelor va fi cel puțin 1:
≥
1
ε
1
−
s
.
{\displaystyle \geq {\frac {1}{\varepsilon ^{1-s))}.}
Rezultă:
H
ε
s
(
X
)
≥
1
ε
1
−
s
{\displaystyle H_{\varepsilon }^{s}(X)\geq {\frac {1}{\varepsilon ^{1-s))))
.Deci:
H
s
(
X
)
=
∞
{\displaystyle H^{s}(X)=\infty \,}
.Pentru
s
=
1
{\displaystyle s=1\,}
: Considerând cele două cazuri anterioare, obținem:
H
1
(
X
)
=
1
{\displaystyle H^{1}(X)=1\,}
.Așadar:
dim
X
=
1
{\displaystyle \dim X=1\,}
.Cercul are dimensiune Hausdorff 1.Dimensiunea Hausdorff a reprezentării triadice Cantor este
ln
2
ln
3
{\displaystyle {\frac {\ln 2}{\ln 3))}
. Dimensiunea Hausdorff a triunghiului lui Sierpinski este
ln
3
ln
2
{\displaystyle {\frac {\ln 3}{\ln 2))}
.
Besicovitch, A.S. - On Linear Sets of Points of Fractional Dimensions , Mathematische Annalen 101 (1929)
Mandelbrot, Benoît - The Fractal Geometry of Nature , Lecture notes in mathematics, W. H. Freeman, 1982. ISBN 0-7167-1186-9 .Spații dimensionale Alte dimensiuni Politopuri și forme Dimensiuni după număr Vezi și
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