GEWMETRIA TWN

FRACTALS

Ask seic III

1.

A = {a}, a ∈ Rd . Apìdeixh. To

H0

to

ApodeÐxte ìti

H 0 (E)

dimH A = dimB A = 0.

eÐnai to pl joc twn shmeÐwn tou

E ⊆ Rd ,

dhlad 

eÐnai to arijmhtikì mètro.

A = {a} èqoume H 0 (A) = 1, epomènwc apo ton orismì thc diˆstashc tou Hausdorff èpetai oti dimH A = 0. Exˆllou, gia to monosÔnolo A = {a} o elˆqistoc arijmìc kÔbwn tou Rd , pleurˆc d pou to kalÔptoun eÐnai Nδ (A) = 1. An

'Ara

logNδ (A) = 0,

gia kˆje

δ>0,

opìte

δ (A) dimB (A) = limδ→0 logN −logδ =

0. Sqìlio:

na apodeÐxoume oti 2. An

dimH (E) ≤ dimB (E) gia E ⊆ Rd , mac arkoÔse dimB ({a}) = 0.

Lìgw tou ìti

A ⊆ B =⇒ dimA ≤ dimB

(ìpou

dim : dimH

 

dimB ).

Apìdeixh.

•

t ≥ 0 ja èqoume H t (A) ≤ H t (B). 'Ara {t ≥ 0 : H t (B) = 0} ⊆ {t ≥ 0 : H t (A) = 0}

Gia

kai

dimH (A) = inf {t ≥ 0 : H t (A) = 0} ≤ inf {t ≥ 0 : H t (B) = 0} = dimH (B). •

Efoson

A ⊆ B

ja èqoume

Nδ (A) ≤ Nδ (B),

ìpou

Nδ (·)

eÐnai o

elˆqistoc arijmìc kÔbwn pleurˆc d pou kalÔptoun èna sÔnolo. 'Ara

3. 'Estw

dimB (A) = limδ→0+

E, F ⊆ Rd

kai

logNδ (A) −logδ

f :E→F

≤ limδ→0+

logNδ (B) −logδ

= dimB (B).

apeikìnish tètoia ¸ste

|f (x) − f (y)| ≤ c|x − y|, x, y ∈ E. ApodeÐxte oti

dimf (E) ≤ dimE

(opou

dim : dimH

 

dimB ).

Apìdeixh.

•

Hausdorff. t (f (E)) = δ(f (U )) ≤ cδ(U ), U ⊆ Rn . Epiplèon, Hc P∞ t S∞ inf { i=1 δ (Vi ) : f (E) ⊆ i=1 Vi , δ(Vi ) ≤ c} ≤ P S∞ t t inf { ∞ i=1 δ (f (Ui )) : E ⊆ i=1 Ui , δ(Ui ) ≤ } ≤ cH (E). 'Ara, Gia th diˆstash 'Eqoume oti

H t (f (E)) ≤ cH t (E). Epomènwc,

dimH f (E) = sup{t : H t (f (E)) = +∞} ≤ sup{t : H t (E) = +∞} = dimH E.

•

Gia th diˆstash Ean

Nδ

Box.

tìte o elˆqistoc arijmìc kÔbwn pleurˆc

to

f (E)

eÐnai

Nδc ≤ Nδ .

'Ara,

dimB f (E) = lim 0

δ →0

4.

(aþ)

S∞

dimH (

n=1 En )

logNδ logNδ0 ≤ lim = dimB E. −logδ 0 δ→0 −logδ

= sup{dimH En : n ∈ N }.

(bþ) An eÐnai arijm simo sÔnolo, tìte (gþ)

δ pou kalÔptei to δ 0 = δc pou kalÔptei

eÐnai o elˆqistoc arijmìc kÔbwn pleurˆc

E,

dimH A = 0 kai H s (A) = 0, ∀s>0.

dimH Q = 0.

Apìdeixh.

(a')

S∞

dimH En ≤ dimH (

i=1 En ), ∀n

∈ N

(apo ˆskhsh 1).

'Ara

sup{dimH En : n ∈ N } ≤ dimH (

∞ [

En )

(1)

i=1

sup{dimH En : n ∈ N }0, H s (A) = 0 (Apo ton orismì thc dimH ).

T¸ra,

(g') 5.

afoÔ

dimH Q = 0,

(aþ) 'Estw

apo to (b) diìti to

A = [a, b]d , a0, a ∈ Rd ). ApodeÐxte oti dimH Sd = dimB Sd = d.

(bþ) 'Estw

Rd

èpetai oti

dimH Rd = d.

(gþ)

b−a n pou logNδ (A) limδ→0 −logδ = d

Apìdeixh. (a') O elˆqistoc arijmìc kÔbwn pleurˆc ton

A=

[a, b]d

eÐnai

nd .

'Ara

dimB A =

H d (A) = cd λd (A) ∈ (0, +∞),

Epeid 

èqoume oti

δ=

kalÔptei

dimH A = d.

upˆrqei kÔboc A me Sd ⊆ A. 'Ara dimB (Sd ) ≤ H d (Sd ) ∈ (0, +∞) èqoume oti dimH Sd = d. Apì ta parapˆnw sumperaÐnoume oti d = dimH Sd ≤ dimB Sd ≤ d. Dhlad  dimH Sd = dimB Sd = d.

(b') Gia th sfaÐra

dimB (A) = d.

Rd =

(g')

Sd

Epeid 

S∞

n=1 SQ (0, n),

ˆra

dimH Rd = sup{dimH S(0, n)} = d

(apì

thn ˆskhsh 3 kai to (b') ). 6. 'Estw

A ⊆ Rd ,

sunektikì Apìdeixh.

me

dimH A = s ∈ (0, 1).

Tìte to

A

eÐnai olikˆ mh

1. 'Estw

B ⊆ A, B

sunektikì kai

x0 , y0 ∈ B

me

x0 6= y0 .

OrÐzoume th sunˆrthsh

f : B −→ [0, +∞], me

f (x) = |x − x0 |.

Tìte,

f

|f (x) − f (y)| = ||x − x0 | − |y − y0 || ≤ |x − y|.

Epomènwc, afoÔ h

ikanopoieÐ tic upojèseic thc 'Askhshc 3, èqoume

H s (f (B)) ≤ H s (B) =⇒ dimH (f (B)) ≤ dimH B ≤ s = dimH A0.

7. 'Estw

sÔnolo me

s = dimB A>0

kai

limδ→0+ Nδ δ t ∈

ApodeÐxte oti 0

dimB A = inf {t : limδ→0+ Nδ δ t = 0} = sup{t0 : limδ→0+ Nδ δ t = +∞}. Genikˆ

:

dimB A = inf {t : limsupNδ δ t = 0} = sup{s : limsupNδ δ s = +∞} dimB A = inf {t0 : liminf Nδ δ t = 0} = sup{s0 : liminf Nδ δ s0 = +∞}. Apìdeixh. H sunˆsthsh wc proc

φ(t, δ) = Nδ (A)δ t , t> 0, 00 t : dimB A, dimB BdimH (Q ∩ [0, 1]) = 0,

(apì

(b) kai ˆskhsh 4).

Q ∩ [0, 1] = {qn : n ∈ N } {q }) = 1 . n n=1

(d) 'Oqi. Gia parˆdeigma, en¸

S∞

dimB (

me

dimB ({qn }) = 0,

b 1), τ >0 kai s = dimB E>0. E ⊆ Rd fragmèno, E +τ =: E +τ S(0, logλd (E+τ ) ApodeÐxte oti dimB E = d − limτ →0+ , ìpou λd eÐnai to mètro logτ Lebesque. (Orismìc Minkowski).

9. 'Estw

Apìdeixh. aktÐnac aktÐnac

E mporeÐ na kalufjeÐ apo Nδ (E) to pl joc to Eδ mporeÐ na kalufjeÐ apo tic omìkentrec

An to

δ , tìte 2δ . 'Ara

voln (Eδ ) ≤ Nδ (E)cn (2δ)n , ìpou

cn

o ìgkoc thc monadiaÐac sfaÐrac tou

Rn .

sfaÐrec sfaÐrec

Logarijm¸ntac,

logvoln (Eδ ) −logδ

limδ→0

≤

logNδ (E)+nlogδ+log2n cn . −logδ

'Ara

logvoln (Eδ ) ≤ −n + dimB E. − log δ

Sunep¸c,

dimB E ≥ n − limδ→0 AntÐstrofa, an upˆrqoun kèntra sto

E,

Nδ

logvoln (Eδ ) . log δ

(5)

xènec metaxÔ touc sfaÐrec aktÐnac d me

tìte afoÔ autèc perièqontai sto

E

kai

E ⊆ Eδ ,

opìte

Nδ (E)cn δ n ≤ voln (Eδ ). PaÐrnontac logarÐjmouc,

Nδ (E) logcn nlogδ logvoln (Eδ ) + + ≤ −logδ −logδ −logδ −logδ 'Ara,

limδ→0

Nδ (E) logvoln (Eδ ) − n ≤ limδ→0 . −logδ −logδ

Sunep¸c,

logvoln (Eδ ) logδ

(6)

logvoln (Eδ ) . logδ

(7)

dimB E − n ≤ −limδ→0 Ki epeid 

dimB E ≤ dimB E , dimB E ≤ n − limδ→0

Apo (5) kai (6),

dimB E = n − limδ→0

logvoln (Eδ ) . logδ

(8)

EpÐshc, apo (7)

dimB E ≤ n − limδ→0

logvoln (Eδ ) logvoln (Eδ ) ≤ n − limδ→0 . logδ logδ

(9)

Apo (8) kai (9),

dimB E = n − limδ→0 Kai fusikˆ, an

logvoln (Eδ ) . logδ

dimB E = dimB E , tìte dimB E = n−limδ→0

logvoln (Eδ ) . logδ

'Enjeto

SÔnola Apokop c

Endiafèron parousiˆzei h melèth thc

box-diˆstashc

opoÐa èqei afairejeÐ mia akoloujÐa uposunìlwn touc.

sunìlwn apo ta Gia parˆdeigma,

to sÔnolo tou Cantor prokÔptei an apo to [0,1] afairèsw thn akoloujÐa ( 13 , 32 ), ( 19 , 92 ), ( 79 , 89 )...

A = [a, b] fragmèno kleistì diˆsthma sto R, kai A1 , A2 , ... P A me λ1 (A) = ∞ 1 (Ai ), ìpou i=1 λS λ1 (·) eÐnai to m koc enìc diast matoc. Tìte, to E = A − ∞ i=1 Ai eÐnai sumpagèc me mètro Lebesgue mhdèn, kai sumplhrwmatikˆ diast mata ta An .

'Estw, loipìn,

akoloujÐa xènwn uposunìlwn tou

Onomˆzoume to E sÔnolo ai : a1 ≥ a2 ≥ a3 ≥ .... 'Estw

apokop c.

Jètw

E + r = Er := {x ∈ R : |x − y| ≤ r,

Tìte, an

r≤

1 2 a1

kai upˆrqei

ai = |Ai |

kai diatˆssw ta

gia kˆpoio

y ∈ E}.

n : an+1 ≤ 2r ≤ an ,

V (r) := λ1 (E + r) = 2nr + 2r +

∞ X

ai , an+1 ≤ 2r ≤ an .

(10)

n+1 Epiplèon,

∞ X

−1 a−a i (ai − ai+1 ) ≤ (1 − a) an (1 − a),

(11)

i=n diìti to pr¸to mèloc thc anisìthtac eÐnai to kat¸tero ˆjroisma thc sunˆrthshc

x−a ,

apo to 0 mèqri to

en¸ to deÔtero eÐnai to orismèno ˆjroisma thc

x−a

an .

Oi sqèseic (10) kai (11) ja qrhsimeÔsoun sth sunèqeia.

Gia analu-

tikìterh melèth twn sunìlwn apokop c mporeÐte na apeujunjeÐte sto [9]-1997 (selÐda 150 twn shmei¸sewn tou maj matoc).

10. SÔnola apokop c (

cut-out)

[a, b] ⊆ R kai Ii ⊆ [a, b], i ∈ N me Ii anoiktˆ diast mata, xèna ana P dÔo kai i=1 +∞λ1 (Ii ) = λ1 ([a, b]). To sumpagèc sÔnolo E = [a, b] − S+∞ I kaleÐtai sÔnolo apokop c. Estw ai = λ1 (Ii ), me a1 ≥ a2 ≥ ...>0 i=1 i 'Estw

kai

c1 n−γ ≤ an ≤ cn n−γ , n ≥ n0 (γ) gia kˆpoiec stajerèc

c1 , c2 >0.

(aþ)

n limn→+∞ (− loga logn ) = γ>1,

(bþ)

c3 τ

(gþ)

1− γ1

≤ λ1 (E + τ ) ≤ c4 τ

dimB E =

kai ˆra

P∞

c3 , c4 >0,

logan c1 n−γ ≤ an ≤ cn n−γ , tìte γ ≤ limn→+∞ ( −logn )≤

c1 n−γ ≤ an ,

n=1 c1

(b') An to

gia kˆpoiec stajerèc

logan limn→+∞ ( −logn ) = γ.

T¸ra, epeid  'Etsi,

1− γ1

:

1 γ.

Apìdeixh. (a') AfoÔ

γ

ApodeÐxte oti

(12)

r

n−γ
1

= λ1 (A)< + ∞.

(AfoÔ h seirˆ sugklÐnei).

eÐnai arketˆ mikrì kai

an+1 ≤ 2r