Fractal Geometry: Mathematical Foundations and Applications - Kenneth Falconer

Comprehensive practice flashcards in vocabulary style covering the mathematical foundations, dimension theories, techniques, and applications of fractal geometry based on Kenneth Falconer's textbook.

Middle third Cantor set

The fractal subset of $[0, 1]$ constructed by starting with $E_0 = [0, 1]$ and repeatedly removing the open middle third of each interval at step $k$, formally defined as $F = \bigcap_{k=0}^{\infty} E_k$, with Hausdorff and box dimensions equal to $\frac{\log(2)}{\log(3)} \approx 0.631$.

von Koch curve

The continuous, nowhere-differentiable curve obtained in the limit from a unit interval by repeatedly replacing the middle third of each line segment by the other two sides of an equilateral triangle, possessing infinite length, zero planar area, and Hausdorff dimension $\frac{\log(4)}{\log(3)} \approx 1.262$.

Similarity dimension

The dimension value $s$ assigned to a self-similar set composed of $m$ non-overlapping copies of itself, each scaled down by a factor of $r$, given by $-\frac{\log(m)}{\log(r)} = \frac{\log(m)}{\log(1/r)}$.

Diameter (of a set)

The greatest Euclidean distance between any pair of points in a non-empty subset $U \subset \mathbb{R}^n$, defined mathematically as $|U| = \sup\{|x - y| : x, y \in U\}$.

Compact set (in $\mathbb{R}^n$)

A subset of $\mathbb{R}^n$ for which every open cover admits a finite subcover, which in Euclidean space is equivalent to being both closed and bounded.

Totally disconnected set

A topological set in which the connected component of every point consists solely of that single point, such that no two distinct points belong to the same connected component.

Borel sets

The smallest family of subsets of $\mathbb{R}^n$ containing all open and closed sets that is closed under countable unions, countable intersections, and set complementation.

Isometry (Congruence)

A distance-preserving transformation $S : \mathbb{R}^n \rightarrow \mathbb{R}^n$ satisfying $|S(x) - S(y)| = |x - y|$ for all $x, y \in \mathbb{R}^n$, preserving angles and transforming sets into geometrically congruent sets.

Similarity

A transformation $S : \mathbb{R}^n \rightarrow \mathbb{R}^n$ that multiplies all distances by a constant positive ratio $c > 0$, such that $|S(x) - S(y)| = c|x - y|$ for all $x, y \in \mathbb{R}^n$.

Affine transformation (Affinity)

A mapping $S : \mathbb{R}^n \rightarrow \mathbb{R}^n$ of the form $S(x) = T(x) + a$, where $T$ is a non-singular linear transformation and $a \in \mathbb{R}^n$, which may contract or expand by different ratios in different directions.

Hölder function

A mapping $f : X \rightarrow Y$ between metric spaces satisfying $|f(x) - f(y)| \le c|x - y|^\alpha$ for all $x, y \in X$, where $c \ge 0$ and $\alpha > 0$ is the Hölder exponent.

Lipschitz function

A mapping $f : X \rightarrow Y$ satisfying $|f(x) - f(y)| \le c|x - y|$ for all $x, y \in X$ and some constant $c \ge 0$, representing a Hölder function of exponent $\alpha = 1$.

Bi-Lipschitz function

A bijection $f : X \rightarrow Y$ satisfying $c_1|x - y| \le |f(x) - f(y)| \le c_2|x - y|$ for all $x, y \in X$, with $0 < c_1 \le c_2 < \infty$, ensuring that both $f$ and $f^{-1}$ are Lipschitz.

Measure (on $\mathbb{R}^n$)

A non-negative set function $\mu$ defined on subsets of $\mathbb{R}^n$ such that $\mu(\emptyset) = 0$, $\mu(A) \le \mu(B)$ for $A \subset B$, and $\mu\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mu(A_i)$ whenever $\{A_i\}$ are disjoint Borel sets.

Mass distribution

A measure $\mu$ supported on a bounded subset of $\mathbb{R}^n$ such that the total mass satisfies $0 < \mu(\mathbb{R}^n) < \infty$.

Support of a measure

The smallest closed set $X \subset \mathbb{R}^n$ such that $\mu(\mathbb{R}^n \setminus X) = 0$, denoted by $\text{spt}\,\mu$, which contains all points $x$ where $\mu(B(x, r)) > 0$ for all radii $r > 0$.

Lebesgue measure ($\mathcal{L}^n$)

The standard measure on $\mathbb{R}^n$ generalizing classical length ($n=1$), area ($n=2$), and volume ($n=3$), defined for coordinate parallelepipeds as the product of their side lengths.

$\delta$-Cover

A countable or finite family of sets $\{U_i\}$ whose union contains a given set $F$, with each set having diameter satisfying $0 \le |U_i| \le \delta$.

$s$-Dimensional Hausdorff measure ($\mathcal{H}^s$)

The outer measure on subsets of $\mathbb{R}^n$ defined by $\mathcal{H}^s(F) = \lim_{\delta \to 0} \mathcal{H}_\delta^s(F)$, where $\mathcal{H}_\delta^s(F) = \inf\left\{\sum_{i} |U_i|^s : \{U_i\}\text{ is a }\delta\text{-cover of }F\right\}$.

Hausdorff dimension ($\dim_H F$)

The unique critical exponent $s$ where the $s$-dimensional Hausdorff measure $\mathcal{H}^s(F)$ jumps from $\infty$ to $0$, formally defined as $\dim_H F = \inf\{s \ge 0 : \mathcal{H}^s(F) = 0\} = \sup\{s : \mathcal{H}^s(F) = \infty\}$.

$s$-Set

A Borel set $F \subset \mathbb{R}^n$ of Hausdorff dimension $s$ for which the $s$-dimensional Hausdorff measure is strictly positive and finite, satisfying $0 < \mathcal{H}^s(F) < \infty$.

Box-counting dimension

The dimension of a non-empty bounded set $F \subset \mathbb{R}^n$ defined as $\dim_B F = \lim_{\delta \to 0} \frac{\log(N_\delta(F))}{-\log(\delta)}$ (if the limit exists), where $N_\delta(F)$ is the minimum number of sets of diameter at most $\delta$ needed to cover $F$.

Lower and upper box-counting dimensions

The lower and upper bounds of box dimension defined by $\underline{\dim}_B F = \liminf_{\delta \to 0} \frac{\log(N_\delta(F))}{-\log(\delta)}$ and $\overline{\dim}_B F = \limsup_{\delta \to 0} \frac{\log(N_\delta(F))}{-\log(\delta)}$, describing the limiting logarithmic rates as $\delta \to 0$.

Minkowski content

The limiting content coefficient $c$ when the volume of the $\delta$-neighbourhood $F_\delta$ of a set $F \subset \mathbb{R}^n$ scales as $\text{vol}_n(F_\delta) \sim c \delta^{n-s}$ as $\delta \to 0$.

Packing dimension ($\dim_P F$)

A metric dimension dual to Hausdorff dimension based on packing disjoint balls of varying small radii centred in $F$, which coincides with the modified upper box-counting dimension $\text{dim}_{\text{MB}} F$.

Divider dimension

The dimension of a Jordan curve $C$ defined by $\lim_{\delta \to 0} \frac{\log(M_\delta(C))}{-\log(\delta)}$, where $M_\delta(C)$ is the maximum number of points ordered along $C$ spaced at step distance $\delta$ apart.

Dimension print

A two-parameter fingerprint $\text{print}\,F$ of a planar set $F$ defined as the set of pairs $(s, t)$ for which the rectangle-based Hausdorff-type measure $\mathcal{H}^{s,t}(F) > 0$, capturing directional and anisotropic features.

Mass distribution principle

A foundational theorem stating that if a mass distribution $\mu$ on $F$ satisfies $\mu(U) \le c|U|^s$ for all sets $U$ with $|U| \le \varepsilon$, then $\mathcal{H}^s(F) \ge \frac{\mu(F)}{c} > 0$ and $s \le \dim_H F \le \underline{\dim}_B F \le \overline{\dim}_B F$.

$s$-Energy ($I_s(\mu)$)

The potential-theoretic integral of a mass distribution $\mu$ given by $I_s(\mu) = \iint \frac{d\mu(x)\,d\mu(y)}{|x - y|^s}$, whose finiteness implies that $\mathcal{H}^s(F) = \infty$ and $\dim_H F \ge s$.

Frostman's lemma

A theorem establishing that if $F \subset \mathbb{R}^n$ is a Borel set with $0 < \mathcal{H}^s(F) \le \infty$, there exists a compact set $E \subset F$ with $0 < \mathcal{H}^s(E) < \infty$ and a constant $b$ such that $\mathcal{H}^s(E \cap B(x, r)) \le b r^s$ for all $x \in \mathbb{R}^n$ and $r > 0$.

Densities of an $s$-set

The lower and upper limiting concentrations of an $s$-set $F$ at $x$ defined by $\underline{D}^s(F, x) = \liminf_{r \to 0} \frac{\mathcal{H}^s(F \cap B(x, r))}{(2r)^s}$ and $\overline{D}^s(F, x) = \limsup_{r \to 0} \frac{\mathcal{H}^s(F \cap B(x, r))}{(2r)^s}$.

Regular point (of an $s$-set)

A point $x$ of an $s$-set $F$ at which the lower and upper densities are both equal to 1, such that $\underline{D}^s(F, x) = \overline{D}^s(F, x) = 1$.

Regular and irregular $s$-sets

An $s$-set $F$ is regular if $\mathcal{H}^s$-almost all of its points are regular, and irregular if $\mathcal{H}^s$-almost all of its points are irregular; any $s$-set in $\mathbb{R}^2$ with non-integral $s$ is necessarily irregular.

Rectifiable curve

A non-self-intersecting Jordan curve $C \subset \mathbb{R}^2$ of positive, finite total polygonal length, which satisfies $\mathcal{H}^1(C) = L(C) < \infty$ and forms a regular 1-set.

Curve-like 1-set

A 1-set $F \subset \mathbb{R}^2$ that is contained in a countable union of rectifiable curves, making it a regular 1-set with lower density equal to 1 almost everywhere.

Curve-free 1-set

A 1-set $F \subset \mathbb{R}^2$ whose intersection with every rectifiable curve has $\mathcal{H}^1$-measure zero, which completely characterizes irregular 1-sets in the plane.

Tangent to an $s$-set

A directional orientation $\theta$ at $x$ for an $s$-set $F$ defined by $D^s(F, x) > 0$ and $\lim_{r \to 0} r^{-s}\mathcal{H}^s(F \cap (B(x, r) \setminus S(x, \theta, \phi))) = 0$ for every angular sector width $\phi > 0$.

Projection theorem (Marstrand)

A foundational theorem stating that for a Borel set $F \subset \mathbb{R}^2$, the orthogonal projections $\text{proj}_\theta F$ satisfy $\dim_H(\text{proj}_\theta F) = \dim_H F$ for almost all $\theta \in [0, \pi)$ if $\dim_H F \le 1$, and have positive length (dimension 1) for almost all $\theta$ if $\dim_H F > 1$.

Product formula for Hausdorff dimension

The inequality $\dim_H(E \times F) \ge \dim_H E + \dim_H F$ holding for arbitrary Borel sets $E \subset \mathbb{R}^n$ and $F \subset \mathbb{R}^m$, with equality occurring whenever $\dim_H F = \overline{\dim}_B F$.

Class $\mathcal{C}_s$ (Large intersection sets)

A class of subsets of $\mathbb{R}$ of Hausdorff dimension at least $s$ containing lim sup sets $F \supset \limsup_{k \to \infty} E_k$ with $\lim_{k \to \infty} \mathcal{H}^s_\infty(I \cap E_k) = |I|^s$, whose countable intersections still have Hausdorff dimension at least $s$.

Iterated function system (IFS)

A finite family of contraction mappings $\{S_1, S_2, \dots, S_m\}$ on a closed domain $D \subset \mathbb{R}^n$ with ratios $c_i < 1$, which uniquely determines a compact invariant set $F = \bigcup_{i=1}^m S_i(F)$.

Hausdorff metric

The metric on non-empty compact subsets of a metric space defined by $d(A, B) = \inf\{\delta : A \subset B_\delta \text{ and } B \subset A_\delta\}$, under which the space of compact subsets is complete.

Open set condition (OSC)

A geometric separation condition for an IFS of similarities requiring a non-empty bounded open set $V$ such that $V \supset \bigcup_{i=1}^m S_i(V)$, with the union on the right being pairwise disjoint.

Self-affine set

The invariant attractor of an iterated function system whose mappings are affine transformations of the form $S_i(x) = T_i(x) + b_i$, where contractions generally contract space at different rates in different directions.

Singular value function ($\phi^s(T)$)

A submultiplicative function of a linear contraction $T$ with singular values $\alpha_1 \ge \alpha_2 \ge \dots \ge \alpha_n$ defined by $\phi^s(T) = \alpha_1 \alpha_2 \dots \alpha_{r-1} \alpha_r^{s-r+1}$, where $r-1 < s \le r$, used to calculate the dimension of self-affine sets.

Collage theorem

A bound on the distance between a target compact set $E$ and the invariant attractor $F$ of an IFS with contraction ratio $c < 1$, stating that $d(E, F) \le \frac{d\left(E, \bigcup_{i=1}^m S_i(E)\right)}{1 - c}$.

Normal number (to base $m$)

A real number in $[0, 1)$ whose base-$m$ expansion contains each digit $0, 1, \dots, m-1$ in equal asymptotic proportions of $\frac{1}{m}$; the set of such numbers has Lebesgue measure 1.

$\alpha$-Well approximable number

A real number $x$ satisfying the Diophantine approximation inequality $\left\|q x\right\| \le q^{1-\alpha}$ for infinitely many positive integers $q$, where $\left\|\cdot\right\|$ denotes distance to the nearest integer.

Jarník's theorem

A classical theorem in metric number theory proving that for $\alpha > 2$, the set of $\alpha$-well approximable numbers in $[0, 1]$ has Hausdorff dimension exactly equal to $\frac{2}{\alpha}$.

Weierstrass function

The continuous, nowhere-differentiable fractal function $f(t) = \sum_{k=1}^{\infty} \lambda^{(s-2)k} \sin(\lambda^k t)$ with $\lambda > 1$ and $1 < s < 2$, whose graph has box dimension equal to $s$ for sufficiently large $\lambda$.

Autocorrelation function ($C(h)$)

The long-term average correlation of a time-dependent signal $f$ with itself across a time displacement $h$, defined as $C(h) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T (f(t + h) - \bar{f})(f(t) - \bar{f})\,dt$.

Besicovitch set (Kakeya problem)

A planar set containing a unit straight-line segment in every possible direction, which can be constructed with planar Lebesgue measure (area) equal to zero, yet must have Hausdorff dimension 2.

Removable set (Complex analysis)

A compact set $F \subset \mathbb{C}$ such that any bounded analytic function defined on $V \setminus F$ (with open $V \supset F$) extends analytically to the entire domain $V$.

Convex function

A continuous function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ satisfying $f(\lambda x + (1 - \lambda)y) \le \lambda f(x) + (1 - \lambda)f(y)$ for all $0 \le \lambda \le 1$, whose set of non-differentiable points has Hausdorff dimension at most 1.

Discrete dynamical system

An iterative system defined by a continuous map $f : D \rightarrow D$ generating discrete trajectories or orbits $\{x, f(x), f^2(x), \dots\}$ from an initial state $x \in D$.

Attractor (Dynamical systems)

A minimal, closed, invariant set $F$ ($f(F) = F$) towards which all orbits starting in an open neighbourhood $V$ (the basin of attraction) converge under iteration.

Repeller

A closed invariant set in a dynamical system from which all nearby points not in the set are iterated away over time, acting as an attractor for the inverse mapping $f^{-1}$.

Chaos (Dynamical systems)

Dynamical action on a set $F$ characterized by the existence of a dense orbit in $F$, dense periodic points in $F$, and sensitive dependence on initial conditions.

Logistic map

The polynomial family $f_\lambda(x) = \lambda x(1 - x)$ on $[0, 1]$ exhibiting period-doubling bifurcations that accumulate at $\lambda_\infty \approx 3.570$, where the attractor is a Cantor-like fractal set.

Feigenbaum constant

The universal geometric scaling ratio $\delta \approx 4.6692$ governing the rate at which successive period-doubling bifurcations occur in parameter space for smooth unimodal mappings.

Baker's transformation

A discontinuous stretching and cutting mapping on the unit square $[0, 1]^2$ given by $f(x, y) = (2x, \lambda y)$ for $0 \le x \le \frac{1}{2}$ and $f(x, y) = (2x - 1, \lambda y + \frac{1}{2})$ for $\frac{1}{2} < x \le 1$, possessing a chaotic attractor of dimension $1 + \frac{\log(2)}{-\log(\lambda)}$.

Hénon map

The planar quadratic transformation $f(x, y) = (y + 1 - a x^2, b x)$ that contracts area at a constant rate $-b$, possessing a strange attractor of dimension $\approx 1.26$ when $a = 1.4$ and $b = 0.3$.

Solenoid

The fibrous attractor in a solid torus $D$ generated by repeatedly stretching, thinning, and wrapping the torus twice within itself, with cross-sections intersecting the attractor in a Cantor-like set.

Lorenz attractor

The strange fractal attractor of Hausdorff dimension $\approx 2.06$ arising from the 3-dimensional system of autonomous differential equations modelling thermal convection in a fluid layer heated from below.

Liapounov exponents

The average logarithmic rates of exponential stretching or contraction $\lambda_i = \lim_{k \to \infty} \frac{1}{k} \log(a_k(x))$ along the orthogonal axes of an infinitesimal ellipsoid under iteration of a dynamical system.

Filled-in Julia set ($K(f)$)

The set of points $z \in \mathbb{C}$ whose orbits under repeated iteration of a polynomial $f$ remain bounded, defined as $K(f) = \{z \in \mathbb{C} : f^k(z) \not\to \infty\text{ as } k \to \infty\}$.

Julia set ($J(f)$)

The topological boundary of the filled-in Julia set, $J(f) = \partial K(f)$, which is a non-empty, compact, completely invariant set on which the family of iterates $(f^k)$ fails to be normal.

Fatou set (Stable set)

The complement of the Julia set in the complex plane, comprising the open region of points where the family of iterates $(f^k)$ is normal and the dynamics are stable.

Mandelbrot set ($M$)

The set of parameters $c \in \mathbb{C}$ for which the Julia set $J(f_c)$ of the quadratic polynomial $f_c(z) = z^2 + c$ is connected, equivalently defined by $M = \{c \in \mathbb{C} : f_c^k(0) \not\to \infty\text{ as } k \to \infty\}$.

Quasi-circle

A simple closed curve in $\mathbb{C}$ of Hausdorff dimension $s$ that is quasi-self-similar, with any two quasi-circles being bi-Lipschitz equivalent if and only if they share identical Hausdorff dimensions.

Statistically self-similar set

A random fractal whose small magnified subsets have the same statistical probability distribution as the entire set, extending strict self-similarity to stochastic constructions.

Fractal percolation

A recursive random spatial process where a square is subdivided into 9 equal squares, each kept independently with probability $p$, displaying a critical threshold $p_c$ above which opposite sides connect with positive probability.

Brownian motion (Wiener process)

A continuous Gaussian stochastic process $X(t)$ with independent, stationary increments such that $X(t + h) - X(t)$ is normally distributed with mean 0 and variance $h$, producing trails in $\mathbb{R}^n$ ($n \ge 2$) of Hausdorff dimension 2.

Fractional Brownian motion of index-$\alpha$

A continuous Gaussian random process with stationary increments distributed normally with mean 0 and variance $h^{2\alpha}$ ($0 < \alpha < 1$), whose sample graphs have Hausdorff and box dimensions equal to $2 - \alpha$.

Fractional Brownian surface of index-$\alpha$

A 2-dimensional Gaussian random field $X(x, y)$ whose height differences have variance proportional to $|(h, k)|^{2\alpha}$, producing random surfaces of Hausdorff dimension $3 - \alpha$ and contours of dimension $2 - \alpha$.

Multifractal measure

A mass distribution exhibiting complex local concentrations such that the sets of points with local power-law scaling $\mu(B(x, r)) \sim r^\alpha$ form different fractals of varying dimensions for different values of $\alpha$.

Coarse multifractal spectrum ($f_C(\alpha)$)

The spectrum defined by $f_C(\alpha) = \lim_{\varepsilon \to 0} \lim_{r \to 0} \frac{\log^+(N_r(\alpha + \varepsilon) - N_r(\alpha - \varepsilon))}{-\log(r)}$, indicating the power-law growth rate of $r$-mesh cubes where $\mu(C) \approx r^\alpha$.

Legendre spectrum ($f_L(\alpha)$)

The multifractal spectrum obtained by taking the Legendre transform of the moment scaling function $\beta(q)$, defined as $f_L(\alpha) = \inf_{-\infty < q < \infty} \{\beta(q) + \alpha q\}$.

Local dimension (Hölder exponent of a measure)

The logarithmic scaling exponent of a measure $\mu$ at a point $x$, defined as $\dim_{\text{loc}}\mu(x) = \lim_{r \to 0} \frac{\log(\mu(B(x, r)))}{\log(r)}$ whenever the limit exists.

Fine multifractal spectrum ($f_H(\alpha)$)

The Hausdorff singularity spectrum of a measure $\mu$ defined as $f_H(\alpha) = \dim_H F_\alpha$, where $F_\alpha = \{x \in \mathbb{R}^n : \dim_{\text{loc}}\mu(x) = \alpha\}$.

Diffusion-limited aggregation (DLA)

A lattice-based growth simulation where diffusing particles executing random walks adhere upon encountering an existing cluster, generating highly branched fractal aggregates with dimension approximately 1.70.

Sierpiński dipole

A practical multiband fractal antenna formed by two Sierpiński triangles etched vertex-to-vertex, providing sharp resonance peaks at wavelength ratios of $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$ that mirror its similarity ratios.

Trading time ($T(t)$)

A distorted or warped time coordinate $T(t) = \mu[0, t]$ generated by a multifractal measure $\mu$, used in financial models to account for periods of extreme price volatility interspersed with calm intervals.