\(
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Translate}{\operatorname{Translate}}
\newcommand{\Rotate}{\operatorname{Rotate}}
\newcommand{\Scale}{\operatorname{Scale}}
\newcommand{\XYScale}{\operatorname{XYScale}}
\newcommand{\YScale}{\operatorname{YScale}}
\newcommand{\XShear}{\operatorname{XShear}}
\newcommand{\YShear}{\operatorname{YShear}}
\newcommand{\m}[4]{\begin{pmatrix}#1 & #2 \\ #3 & #4\end{pmatrix}}
\newcommand{\K}{\mathcal{K}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\T}{\mathcal{T}}
\newcommand{\e}{\varepsilon}
\newcommand{\diam}{\operatorname{diam}}
\newcommand{\Trace}{\operatorname{Trace}}
\renewcommand{\mid}{\ : \ }
\newcommand{\dim}{\operatorname{dim}}
\newcommand{\Ord}{\operatorname{Ord}}
\newcommand{\Ref}{\operatorname{Ref}}
\newcommand{\O}{\mathcal{O}}
\)

- $f$ has a unique fixed point: there is a single $p\in X$ such that $f(p)=p$;
- given any $q\in X$, the sequence $(x_n)$ defined by $x_0=q$ and $x_{n}=f(x_{n-1})$ for $n>0$ converges to $p.$

Function | Matrix $A$ | Mapping | Action |
---|---|---|---|

$\Scale(s)$ | $\begin{pmatrix} s & 0\\ 0 & s \end{pmatrix}$ | $\begin{bmatrix}x\\ y\end{bmatrix}\mapsto \begin{bmatrix}sx\\ sy\end{bmatrix}$ | Scale a vector by $s$ in the $x$ and $y$ directions. |

$\XYScale(s, t)$ | $\begin{pmatrix} s & 0\\ 0 & t \end{pmatrix}$ | $\begin{bmatrix}x\\ y\end{bmatrix}\mapsto \begin{bmatrix}sx\\ ty\end{bmatrix}$ | Scale a vector by $s$ in the $x$ direction and by $t$ in the $y$ direction. |

$\Rotate(\theta)$ | $\begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix}$ | $\begin{bmatrix}x\\ y\end{bmatrix}\mapsto \begin{bmatrix}x\cos\theta-y\sin\theta\\ x\sin\theta+y\cos\theta\end{bmatrix}$ | Rotate a vector by $\theta$ radians, counter-clockwise around the origin. |

$\XShear(t)$ | $\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$ | $\begin{bmatrix}x\\ y\end{bmatrix}\mapsto \begin{bmatrix}x+ty\\ y\end{bmatrix}$ | Shear a vector in the $x$ direction with a shear parameter $t.$ |

$\YShear(t)$ | $\begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix}$ | $\begin{bmatrix}x\\ y\end{bmatrix}\mapsto \begin{bmatrix}x\\ tx+y\end{bmatrix}$ | Shear a vector in the $y$ direction with a shear parameter $t.$ |

- Let $S_0$ be some compact subset of $\R^2$.
- Choose some very large $N\in\N$.
- For each $n\in\{1,\dots, N\}$, calculate $S_n=F(S_{n-1})=\bigcup_{i=1}^k f_i(S_{n-1})$.
- Return $S_N$.

- Define a discrete probability distribution $P=(p_1,\dots, p_k)$, where $\sum_i p_i = 1$ and $p_i\in(0,1]$.
- Choose some very large $N\in\N$.
- Choose an arbitrary $g_0\in\R^2$.
- For each $n\in\{1,\dots, N\}$:
- Choose a
**random**$j\in\{1,\dots, k\}$ with probability $p_j$. - Calculate $g_n=f_j(g_{n-1})$.

- Choose a
- Return $\{g_0,g_1,\dots,g_N\}$.

- $\dim_\text{ind}(S)=-1$ if and only if $S=\emptyset$.
- $\dim_\text{ind}(S)\leq n$, where $n\in\{0, 1, 2, \dots\}$, if for every point $x\in S$ and each neighborhood $V \subset S$ of the point $x$ there exists an open set $U \subset S$ such that $x\in U\subset V$ and $\dim_\text{ind}(\partial U)\leq n - 1$.
- $\dim_\text{ind}(S)=n$ if $\dim_\text{ind}(S)\leq n$ and $\dim_\text{ind}(S)>n-1$, i.e., the inequality $\dim_\text{ind}(S)\leq n-1$ does not hold.
- $\dim_\text{ind}(S)=\infty$ if $\dim_\text{ind}(S)>n$ for all $n\in\{-1,0,1\dots\}$.

- $\dim_\text{Ind}(S)=-1$ if and only if $S=\emptyset$.
- $\dim_\text{Ind}(S)\leq n$, where $n\in\{0, 1, 2, \dots\}$, if for every closed set $A\subset S$ and each open set $V\subset S$ which contains the set $A$ there exists an open set $U\subset X$ such that $A\subset U\subset V$ and $\dim_\text{Ind}(\partial U)\leq n - 1$.
- $\dim_\text{Ind}(S)=n$ if $\dim_\text{Ind}(S)\leq n$ and $\dim_\text{Ind}(S)>n-1$.
- $\dim_\text{ind}(S)=\infty$ if $\dim_\text{Ind}(S)>n$ for all $n\in\{-1,0,1\dots\}$.

- $\dim_{LCD}(S)\leq n$, where $n\in\{-1,0,1,\dots\}$, if for every finite open cover of the space $S$ has a finite open refinement of order at most $n$.
- $\dim_\text{LCD}(S)=n$ if $\dim_\text{LCD}(S)\leq n$ and $\dim_\text{LCD}(S)>n-1$.
- $\dim_\text{LCD}(S)=\infty$ if $\dim_\text{LCD}(S)>n$ for all $n\in\{-1,0,1,\dots\}$.

- every finite open cover of $S$ has a refinement with order $n$ or less, and
- there is some open cover with no refinement of order $n - 1$ or less;
- sufficiently, there is no open cover of order $n - 1$ or less.

- $F(U)\subseteq U$, and
- $f_i(U)\cap f_j(U)=\emptyset$ if $i\neq j$.