Fractal Sets: Dynamical, Dimensional and Topological Properties

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IN

DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS

,

STOCKHOLM SWEDEN 2018

Fractal Sets: Dynamical,

Dimensional and Topological

Properties

NANCY WANG

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INOM EXAMENSARBETE TEKNIK, GRUNDNIVÅ, 15 HP , STOCKHOLM SVERIGE 2018

Fraktalmängder: Dynamiska,

Dimensionella och Topologiska

Egenskaper

NANCY WANG

KTH

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aij œ {0, 2} b= 0.b1b2b3b4. . . bj = Y ] [ 0 aij = 2 2 aij = 0 b 0 2 bj b _{”= f}≠1(k) _k = 1, 2, . . . , n ≠ 1 b= f≠1(n) bn ann n f≠1 1 3 1 33 1 3 + 2312 + 22 1 33 + · · · = Œ ÿ i=0 2i 1 3i+1 = 1 3 Œ ÿ i=0 3₂ 3 4i = 1. C I _{œ C} I L(I) I I L(I) > 0 I _{œ C}

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Vn n n= lnN(‘) ≠ lnVn ln11 ‘ 2 . n = lim ‘æ0 lnN(‘) ≠ lnVn ln11 ‘ 2 = {Vn } = lim ‘æ0 lnN(‘) ln11 ‘ 2. S Rn S S S _{µ R}n dimB(S) = lim ‘æ0sup lnN(‘) ln11 ‘ 2 . dimB(S) = lim_‘ æ0inf lnN(‘) ln11 ‘ 2 .

dimB(S) = dimB(S) dimB(S)

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S1 µ S2 H–(S1) Æ H–(S2) H–(ﬁŒ i=1Si) Æ qŒi=1H–(Si) {Si} Rd d(S1, S2) Ø 0 H–(S1ﬁ S2) = H–(S1) + H–(S2) H–(S) < Œ _{— > –} _H—(S) = 0 _H–(S) > 0 — < – _H—(S) = Œ H–(S) = lim_” æ0H – ”(S) = lim_” æ0inf I ÿ i (diam Ui)– : {Ui} ” S J . – < — 0 Æ diam Ui Æ ”

(diam Ui)— = (diam Ui)—≠–(diam Ui)– Æ ”—≠–(diam Ui)–.

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Rn _S –= sup{— Ø 0 : H—(S) = Œ} = inf{— Ø 0 : H—(S) = 0} – = dim_H(S) – S ln2 ln3 D _CL 1 = D fl [0,13] C1R = D_{fl [}2₃,1] 1 3 C1 = C1Lfi C1R iii) Hd(C1) = Hd(C1L) + Hd(C1R) = 3₁ 3 4d Hd(C1) + 3₁ 3 4d Hd(C1) = 2 3₁ 3 4d Hd(C1) d 0 < Hd(C 1) < Œ d = dimH(C) d= ln2ln3 ln2 ln3 S µ Rn

dimH(S) Æ dimB(S) Æ dimB(S).

Q ﬂ [0, 1] dimB(A) = 1 dimH(A) = 0.

cl(S) S

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cl(A) = [0, 1] cl(A)

dimB(A) = dimB(cl(A)) = 1.

Ai H0(Ai) = 1

dimH(Ai) = 0. ﬁŒi=1Ai dimH(A) = 0

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