Title: Theory / Unimportant / Math / Fractals
Tags: mid1516974248815 duplicateFromOtherBook fromMath hasSeenConjDef

Name: The Cantor set
Definition: The intersection of the \(C^n\)'s with \(C^n=\bigcup_{i=0}^{3^{n-1} }[\frac{3i}{3^n},\frac{3i+1}{3^n}]\).
Page: xvi
Chapter: 0
Definition2: The set of numbers of the form \((0,2)^\omega\) in base 3
Name2: The middle-third Cantor set

Name: Von Koch curve
Definition: At each stage, the middle third of each interval is replaced by the other two sides of an equilateral triangle.
Page: xviii
Chapter: 0

Name: Sierpiński triangle
Definition: repeatedly removing (inverted) equilateral triangles from an initial equilateral triangle of unit side length
Page: xix
Chapter: 0
Name2: Gasket

Name: Cantor dust
Page: xx
Chapter: 0

Name: Random von Koch curve
Definition: Start with a unit line.
At each step, cut a segment in three.
Use the middle segment as a side of an equilateral triangle. Toss a coin to decide on which side is the triangle. Remove the middle segment.
Page: xx
Chapter: 0

Name: Fractal
Definition: has a fine structure, that is, detail on arbitrarily small scales.
Page: xx, xxv
Chapter: 0
Definition2: is too irregular to be described in traditional geometrical language, both locally and globally.
Conjdef: A set which
Definition3: has some form of self-similarity, perhaps approximate or statistical.
Intuition: objects that were too irregular to fit into a traditional geometrical setting
Definition4: its fractal dimension is greater than its topological dimension
Definition5: defined in a very simple way, perhaps recursively.
Etymology: Fractus: broken

Name: The similarity dimension of \(X\)
Definition: \(-\log m/\log r\) when \(X\) is composed of \(m\) copies of itself, scaled by a factor \(r\)
Page: xxiv
Chapter: 0
Variables: \(X\) a geometrical set

Question: The similarity dimension of a line is
Answer: 1 (it is 4 copies of itself, scaled 1/4th time)
Page: xxiv
Chapter: 0
Variables: \(X\) a geometrical set

Question: The similarity dimension of a square is
Answer: 2 (it is 4 copies of itself, scaled 1/2th time)
Page: xxiv
Chapter: 0
Variables: \(X\) a geometrical set

Question: The similarity dimension of a von Koch curve is
Answer: \(\log2/\log 3\) (it is 4 copies of itself, scaled 1/3th time, hence \(-\log4/\log{\frac 19}\))
Page: xxiv
Chapter: 0
Variables: \(X\) a geometrical set

Name: The topological dimension of \(X\)
Definition: An integer \(i\) such that each element of \(X\) has an arbitrarily sall neighbourhoods with boundary of dimension \(i\)
Page: xxiv
Chapter: 0
Variables: \(X\) a geometrical set

Name: A fractal for Mandelbrot
Definition: Object whose Hausdorff dimension is strictly greater than its topological dimension
Page: xxiv
Chapter: 0

Question: Omitted:
Answer: Basic set theory, Lebesgue measure
Page: , 1-13
Chapter: 0

Name: A way to assign a mass measure to each subset of \(E\)
Definition: \(\mathcal E_0=\{E\}\)
Page: 13
Chapter: 1
Variables: \(E\) a set
Definition2: for each \(i>0\), each \(S\in\mathcal E_i\) is contained in some \(S'\in\mathcal E_{i-1}\)
Conjdef: A sequence \((E_i)_{k\in\mathbb N^{>0}}\) such that
Definition3: each \(\mathcal E_k\) is finite
Definition4: each \(\mathcal E_k\) contains disjoint sets
Denoted by: \((\mathcal E_k)_{k\in\mathbb N}\)

Definition: \(\bigcup_{S\in \mathcal E_k}S\)
Page: 13
Chapter: 1
Variables: \(E\) a set
\((\mathcal E_k)_{k\in\mathbb N}\)
\(k\in\mathbb N\)
Notation: \(E_k\)

Name: A masses distribution on \((\mathcal E_k)_{k\in\mathbb N}\)
Definition: given \(S\in\mathcal E_k\), \(\sum_{S'\subseteq S, S'\in\mathcal E_{k+1}}f(S')=f(S)\)
Page: 13
Chapter: 1
Variables: \(E\) a set
\((\mathcal E_k)_{k\in\mathbb N}\), satisfying the
Definition2: sending \(S\) to 0 when \(S\) is disjoint from some \(E_k\)
Conjdef: A function \(f:\mathcal E\to\mathbb R^{>0}\) such that:
Denoted by: \(\mu\)

Definition: the elements of each \(\mathcal E_k\)
Page: 13
Chapter: 1
Variables: \(E\) a set
\((\mathcal E_k)_{k\in\mathbb N}\)
Notation: \(\mathcal E\)
Definition2: the sets disjoint from some \(E_k\)
Conjdef: A set containing

Name: The extension of \(\mu\) on \(\mathbb R^n\)
Definition: The function sending \(A\subseteq\mathbb R^n \) to the infimum of the \(\sum_{i=1}^\infty\mu(U_i)\) such that \(A\cap E_{\infty}\subseteq \bigcup_i U_i\), and \(U_i\in\mathcal E\)
Page: 14
Chapter: 1
Variables: \(E\mathbb R^n\) a set
\((\mathcal E_k)_{k\in\mathbb N}\)
\(\mu\) a measure on \(\mathcal E\)

Name: The support of \(\mu\)
Definition: \(\bigcap_{k}\overline{E_k}\)
Page: 14
Chapter: 1
Variables: \(E\mathbb R^n\) a set
\((\mathcal E_k)_{k\in\mathbb N}\)
\(\mu\) a measure on \(\mathcal E\)
Notation: \(E_{\infty}\)

Name: A Hölder function of exponent \(\alpha\)
Definition: A function \(f:X\to Y\) such that [$]\abs{f(x)-f(y)}\le c\abs{x-y}^\alpha[/$] for some \(c\ge0\)
Page: 7
Chapter: 1
Section: 2
Variables: \(T,S\) metric set,
\(\alpha\in\mathbb R_{>0}\)

Name: A Lipschitz function
Definition: A Hölder function of exponent 1
Page: 7
Chapter: 1
Section: 2
Variables: \(T,S\) metric set,
\(\alpha\in\mathbb R_{>0}\)

Name: A bi-Lipschitz function
Definition: A function \(f:S\to T\) such that [$]c_1\abs{x-y}\le\abs{f(x)-f(y)}\le c_2\abs{x-y}[/$] for some \(0<c_1<c_2\)
Page: 7
Chapter: 1
Section: 2
Variables: \(T,S\) metric set,
\(\alpha\in\mathbb R_{>0}\)

Name: A coordinate parallelepiped
Definition: Some set \(\prod_{i=1}^n[a_i,b_i]\) for some \((a_i\le b_i)_{i=1}^n\)
Page: 12
Chapter: 1
Section: 3
Variables: \(n\in\mathbb N\)
Extra: interval

Name: The Lebesgue measure in \(\mathbb R^n\)
Definition: Function sending \(A\subseteq\mathbb R^n\) to the infimum \(\left\{\sum_{i=1}^\infty\text{vol}^n(A_i)\mid A\subseteq\bigcup_{i=1}^\infty A_i\right\}\) with \(A_i\) coordinate parallelepiped
Page: 12
Chapter: 1
Section: 3
Variables: \(n\in\mathbb N\)
Notation: \(\mathcal L^n\)

Name: A \(\delta\)-cover of \(F\)
Definition: A countable cover \((U_i)\) of \(F\), such that [$]\abs{U_i}\le\delta[/$] for each \(i\), with [$]\abs{U_i}[/$] the diameter of \(U_i\)
Page: 27
Variables: \(n\in\mathbb N\)
\(F\subseteq \mathbb R^n\)
\(\delta>0\)

Definition: \(\lim_{\delta\to 0}\mathcal H^s_\delta\)
Page: 27
Section: 1
Chapter: 2
Variables: \(n\in\mathbb N\)
\(F\subseteq \mathbb R^n\)
\(\delta,s>0\)
Notation: \(\mathcal H^s\)
Name: \(s\)-dimensional Hausdorff measure
Monotonic: in \(F\)
Domain: subset of \(\mathbb R^n\)
Codomain: \(\mathbb R_{\ge0}\cup\{\infty\}\)
Measure: yes

Definition: sending \(F\) to the infimum of [$]\sum_{i}\abs{U_i}^s[/$], for \(\{U_i\}\) a \(\delta\)-cover of \(F\)
Page: 27
Section: 1
Chapter: 2
Variables: \(n\in\mathbb N\)
\(F\subseteq\mathbb R^n\)
\(\delta,s>0\)
Notation: \(\mathcal H_\delta^s\)
Monotonic: in \(F\)
inverse in \(\delta\)
Domain: subset of \(\mathbb R^n\)
Codomain: \(\mathbb R_{\ge0}\cup\{\infty\}\)

Definition: \(\frac{\pi^{\frac n2}}{2^n\left(\frac n2\right)!}\) if \(n\) even
Page: 28
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
Context: Measure theory
Notation: \(c_n\)
Definition2: \(\frac{\pi^{\frac {n-1}2}\left(\frac {n-1}2\right)!}{n!}\) if \(n\) odd
Intuition: Dimension of the \(n\)-dimentional ball of diameter 1

Smaller0: \(\mathcal H^n(F)\)
Greater0: \(\text{vol}^m(F)\)
Page: 28
Chapter: 2
Section: 1
Variables: \(m\le n\in\mathbb N\)
\(F\subseteq\mathbb R^n\)
Smaller to greater: \(\times c_m=\)
Assuming: \(F\) Borel
Greater to smaller: \(=c_m\times \)
Context: Measure theory
Intuition: \(F\) a smooth \(m\)-dimensional submanifold of \(\mathbb R^n\)

Smaller0: \(\mathcal H^n(F)\)
Greater0: \(\text{vol}^n(F)\)
Page: 28
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
\(F\subseteq\mathbb R^n\)
Smaller to greater: \(\times c_n=\)
Assuming: \(F\) Borel
Greater to smaller: \(=c_n\times\)
Context: Measure theory

Smaller0: \(\mathcal H^0(F)\)
Greater0: the cardinal of \(F\)
Page: 28
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
\(F\subseteq\mathbb R^n\)
Smaller to greater: equals
Context: Hausdorff measure

Smaller0: \(\mathcal H^1(F)\)
Greater0: the length of the curve
Page: 28
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
\(F\subseteq\mathbb R^n\) a curve
Smaller to greater: equals
Assuming: \(F\) is smooth

Definition: \(F\mapsto\overline\lim_{\delta\to 0}\frac{\log N_\delta(F)}{-\log\delta}\)
Page: 27
Section: 1
Chapter: 2
Variables: \(n\in\mathbb N\)
Notation: \(\overline{dim}_B\)
Name: The upper box dimension
Monotonic: Yes
Domain: \(\mathcal P(\mathbb R^n)\)
Codomain: \(\mathbb R_{\ge0}\)

Definition: \(F\mapsto\underline\lim_{\delta\to 0}\frac{\log N_\delta(F)}{-\log\delta}\)
Page: 27, 32
Section: 1, 1
Chapter: 2, 2
Variables: \(n\in\mathbb N\)
\(F\subseteq\mathbb R^n\)
Notation: \(\underline{dim}_B\)
Name: The lower box dimension of \(F\)
Index: , 4
Kind: , prop
Monotonic: Yes
Domain: \(\mathcal P(\mathbb R^n)\)
Codomain: \(\mathbb R_{\ge0}\)

Definition: Lower and upper box dimensions if they are equal
Page: 27
Section: 1
Chapter: 2
Variables: \(n\in\mathbb N\)
\(F\subseteq\mathbb R^n\)
Notation: \(\mathtt{dim}_B\)
Name: The box dimension of \(F\)
Monotonic: Yes
Domain: \(\mathcal P(\mathbb R^n)\)
Codomain: \(\mathbb R_{\ge0}\)
Applied to: the middle third cantor set
Example: \(\frac{\log2}{\log3}\)
Applied to2: the Sierpiński triangle,
Example2: \(\frac{\log3}{\log2}\)
Applied to3: Hyper cube
Example3: \(n\)

Definition: \(F\mapsto n-\underline{\lim}_{\delta\to0}\frac{\log\mathcal L^n(F_\delta)}{\log\delta}\)
Page: 27
Section: 1
Chapter: 2
Variables: \(n\in\mathbb N\)
\(F\subseteq\mathbb R^n\)
Notation: \(\overline{dim}_B\)
Name: The upper Minkowski(-Bouligand) dimension of \(F\)
Monotonic: Yes
Domain: \(\mathcal P(\mathbb R^n)\)
Codomain: \(\mathbb R_{\ge0}\)

Definition: \(n-\overline{\lim}_{\delta\to0}\frac{\log\mathcal L^n(F_\delta)}{\log\delta}\)
Page: 27, 32
Section: 1, 1
Chapter: 2, 2
Variables: \(n\in\mathbb N\)
\(F\subseteq\mathbb R^n\)
Notation: \(\underline{dim}_B\)
Name: The lower Minkowski(-Bouligand) dimension of \(F\)
Index: , 4
Kind: , prop
Monotonic: Yes
Domain: \(\mathcal P(\mathbb R^n)\)
Codomain: \(\mathbb R_{\ge0}\)

Definition: Upper and lower Minkowski dimension of \(F\) if they are equal
Page: 27
Section: 1
Chapter: 2
Variables: \(n\in\mathbb N\)
\(F\subseteq\mathbb R^n\)
Notation: \(\mathtt{dim}_B\)
Name: The Minkowski(-Bouligand) dimension of \(F\)
Monotonic: Yes
Domain: \(\mathcal P(\mathbb R^n)\)
Codomain: \(\mathbb R_{\ge0}\)
Measure: not

Definition: the least number of set of diameter \(\delta\) covering \(F\)
Page: 26
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
Context: Fractal
Notation: \(N_\delta\)
Definition2: the smallest number of cubes of size \(\delta\) covering \(F\)
Definition3: largest number of disjoint ball of radius \(\delta\) with centres in \(F\)
Definition4: the number of \(\delta\)-box of the \(\delta\)-mesh which intersects \(F\)
Definition5: the smallest number of balls of radius \(\delta\) covering \(F\)
DefType: Sending \(F\subseteq\mathbb R^n\)

Name: The \(\delta\)-mesh
Definition: The set of \(\delta\)-box
Page: 28
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
\(\delta>0\)
Name2: The \(\delta\)-grid
Typ: Almost a partition of \(\mathbb R^n\), intersections have measure 0/dimension at most \(n-1\)

Name: A \(\delta\)-box
Definition: A set of the form \(\prod_{i=1}^n[m_i\delta,(m_i+1)\delta]\) for some \((m_1,\dots,m_n)\in\mathbb Z^n\)
Page: 27
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
\(\delta>0\)

Name: The \(\delta\)-neighborhood of \(F\)
Definition: the set of \(x+d\) such that \(x\in F\) and [$]\abs{d}\le\delta[/$]
Page: 33
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
\(\delta>0\)
\(F\subseteq\mathbb R^n\)
Context: Fractal
Notation: \(F_\delta\)
Example: sausage like
Intuition: \(F\), slightly sickened
Applied to: A line

Definition: \(\lim_{\delta\to0}\frac{\mathcal L^n(F_\delta)}{\delta^{n-s}}\)
Page: 32
Section: 1
Chapter: 2
Variables: \(n\in\mathbb N\)
\(\delta,s{\gt}0\)
\(F\subseteq\mathbb R^n\)
Name: The \(s\)-dimensional Minkowski content of \(F\)
Applied to: A \(m\)-dimensional smooth subset
Example: \(m\)

Smaller0: The Minkowski-Bouligand dimension of \(F\)
Greater0: The box dimension of \(F\)
Page: 32
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
\(\delta,s>0\)
\(F\subseteq\mathbb R^n\)
Smaller to greater: equals
Assuming: the Minkowski-Bouligand dimension exists

Smaller0: the lower/upper Minkowski-Bouligand dimension of \(F\)
Greater0: the lower/upper box dimension of \(F\)
Page: 32
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
\(\delta,s>0\)
\(F\subseteq\mathbb R^n\)
Smaller to greater: equals

Question: The box dimension of \(F\)
Answer: Assuming that the size of intervals not in \(F\) is \((l_i)\) then \(-\frac1{\lim_{j\to\infty}\frac{\log l_j}j}\) if it exists
Page: 33
Chapter: 2
Section: 1
Variables: \(F\subseteq\mathbb R\)
Assuming: \(F\) compact
Extra: what's the difference from \(\lim_{j\to\infty}-\frac{\log j}{\log l_j}\)