update readme and report

README.md

@@ -42,6 +42,12 @@ pip install numpy
 ```
 I also use `pathlib`, but that comes built into Python versions >= 3.4, so make sure your Python version includes `pathlib`.
 
+#### For the fractal viewer 
+
+To run the fractal viewer with Python, you'll need : 
+
+`pygame` (versions 2.0.1 and up), and `PyOpenGL`, both installable with pip.
+
 
 ### How to use
 
@@ -65,7 +71,8 @@ You can lerp two images of same shape together using the lerp widget, which yiel
 
 You can save images as `.frctl` files ( a "fractal file" containing metadata, array data and the actual image).
 You can then load `.npy` or `.frctl` files (`npy` files need to be arrays of shape `(2,n)` representing a Taylor expansion of the function of order $n$, separating real and imaginary parts).
-`frctl` files can also be opened outside of the browser, opening them if it's through the executable, but to set this as default, either use the `reg` key generator or set it yourself.
+`frctl` files can also be opened outside of the browser, you can open them using the explorer executable or the browser executable. I recommend setting the explorer executable as the default application.
+*If you want to set the explorer executable as the default application for these in the **Windows** registry, I included a regkey file in this repository where you only have to change `%directory%` to the directory in which the executable is.*
 
 Of course, if you like a function but don't want to save it, just view it in higher resolution (or lower resolution), you can "enhance" a selected image.
 

report/Annex/frctl-art-annex.tex

@@ -0,0 +1,213 @@
+\documentclass{article}
+\usepackage[document]{ragged2e}
+\usepackage{enumitem}
+\usepackage{amssymb}
+\usepackage[spaces,hyphens]{url}
+\usepackage{amsthm}
+\usepackage{bbm}
+\usepackage{dsfont}
+\usepackage{fullpage}
+\usepackage{fancyhdr}
+\usepackage{titlesec}
+\usepackage{hyperref}
+\usepackage{url}
+\usepackage{xcolor}
+\usepackage{stmaryrd}
+\usepackage{tikzpagenodes}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{CJKutf8}
+\usepackage[english]{babel}
+\usepackage{titlesec}
+\usepackage{pgfplots}
+\pgfplotsset{width=10cm,compat=1.9}
+\usepgfplotslibrary{external} 
+\usepackage{tikz,tkz-tab,amsmath}
+\usetikzlibrary{arrows}
+\graphicspath{ {./images/} }
+\newcommand\E{e}
+\newcommand\R{\mathbb{R}}
+\newcommand\N{\mathbb{N}}
+\newcommand\Z{\mathbb{Z}}
+\newcommand\K{\mathbb{K}}
+\newcommand\Q{\mathbb{Q}}
+\newcommand\C{\mathbb{C}}
+\newcommand\T{\mathcal{T}}
+\usepackage[headsep=1cm, headheight=1cm]{geometry}
+% !TeX spellcheck = en_GB 
+
+\newcommand\dspst{\displaystyle}
+\def\prop#1{\underline{\textbf{#1}}}
+
+\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]}
+\let\endchangemargin=\endlist
+
+%\titleformat{\part}[block]{\filcenter}{}(1em){}
+\titleformat{\part}[block]{\Huge\bfseries\filcenter}{\partname{} \\ \thepart}{1em}{\huge}
+\renewcommand{\thesubsection}{\Alph{section}.\arabic{subsection}}
+\renewcommand{\thesubsubsection}{\Alph{section}.\arabic{subsection}.\arabic{subsubsection}}
+
+%Defines emphasis color
+\definecolor{rycolor}{RGB}{200,20,255}
+\definecolor{note}{RGB}{100,100,100}
+
+%Defining a multi-line comment
+\newcommand{\comment}[1]{}
+
+\title{Programming and Algorithms Project \\ Annex - Julia and Fatou sets}
+
+\author{Christopher Mazzerbo}
+\date{Last updated on \today}
+
+
+\begin{document}
+\setcounter{section}{1}
+
+\titlespacing{\subsubsection}{5mm}{0cm}{0cm}
+
+%title
+
+\fancypagestyle{plain}{
+\fancyfoot{}
+}
+\pagestyle{plain}
+\maketitle
+\vspace{5mm}
+\hrule
+\pagebreak
+
+
+%Edit header
+\pagestyle{fancy}
+\fancyhf{}
+\fancyhfoffset[L]{1cm} % left extra length
+\fancyhfoffset[R]{1cm} % right extra length
+\rhead{Programming and Algorithms Project - fractal art}
+\lhead{\bfseries Christopher Mazzerbo}
+\fancyfoot{}
+\fancyfoot[C]{\thepage}
+
+\tableofcontents
+\pagebreak
+
+
+
+In this document we will use the notation $\overline{\C}$ to denote the Riemann sphere : $\overline{\C} = \C \cup \lbrace \infty \rbrace$. \\
+
+\subsection{Definitions}
+
+\prop{Meromorphic function} : \\
+\begin{changemargin}{1cm}{0cm}
+Let $f : \overline{\C} \to \overline{\C}$. We say $f$ is \textit{meromorphic} on an open set $D \subset \overline{\C}$ if $f$ is holomorphic (that is to say, complex-differentiable) on $D \backslash S$ where $S$ is a set of isolated points ($\#S < \infty$). \\
+\vspace{2mm}
+I will write $M(X)$ the space of all complex functions that are meromorphic on $X \subseteq \overline{\C}$.
+\end{changemargin}
+\vspace{5mm}
+
+
+\prop{Julia sets, Fatou sets} : \\
+\begin{changemargin}{1cm}{0cm}
+Let $U$ be an open subset of $\overline{\C}$ and let $f : U \to \overline{\C}$ be a meromorphic function. \\
+$$\mathcal{F}_f = \biggr\lbrace z \in U, \, \, \exists V_z \subset U, \, \, \exists (f^{n_k})_{n_k \in \N}, n_1 < n_2 < \dots, \quad (f^{n_k}) \text{ converges uniformly on compacts subsets of } V_z \biggr\rbrace$$
+Taking $U = \C$ yields what we call the \underline{Fatou set} of $f$, and we call the complement of this set, the \underline{Julia set} \cite{Sut14} section 5. We then have $\C = \mathcal{F}_f \oplus \mathcal{J}_f$. \\
+\vspace{2mm}
+It's important to know that $M(\overline{\C}) \ni f \mapsto \mathcal{J}_f \in \text{Julia sets of }M(\overline{\C})$ is not a bijection. \\
+Two \textit{different} functions can have the same Julia set \cite{Lev97}.
+\end{changemargin}
+\vspace{5mm}
+\pagebreak
+
+\subsection{Two examples}
+
+\subsubsection{Example of $w \in \mathcal{J}_f$ for a very simple $f$}
+
+Consider $f : z \mapsto z^2$, and $w = 1$. \\
+\vspace{5mm}
+We will prove $w \in \mathcal{J}_f$ by finding a sequence $(z_k)_k$ for which iterates of $f$ around any neighbourhood $U$ of $w$ do not have any uniformly converging subsequences on any compact subset of said neighbourhood. \\
+\vspace{5mm}
+\begin{proof}
+Consider $r > 0$, defining a neighbourhood $U$ of $w$ by $U = B(w,3r)$. \\
+\vspace{2mm}
+
+Let's take a sequence $(z_k)_{k \in \N}$ defined by $z_k = w - \frac{r}{k}$. It's easy to see $z_k$ converges to $w$. \\
+\vspace{5mm}
+Now consider $f^n(z_k)$ for a fixed $k$. \\
+We can deduce from basic power properties that : \\
+$$f^n(z_k) \underset{n \to \infty}{\longrightarrow} 0$$
+However : \\
+$$f^n(w) \underset{n \to \infty}{\longrightarrow} 1 \text{ because } f^n(w) = f^n(1) = 1 \text{ for all } n$$
+But then we have $\lVert f^n(z_k) - f^n(w) \rVert \underset{n \to \infty}{\longrightarrow} 1$ for all $k$, which contradicts uniform convergence of $f^n$ or any of its subsequences. \\
+This result is true without having to bring up compactness as the discontinuity of $f^n$'s limit in this neighbourhood is independent of the subset we're studying. \\
+\vspace{2mm}
+Therefore, $w = 1$ is \textbf{not} in the Fatou set. \\
+\vspace{5mm}
+We know that $\overline{\C} = \mathcal{F}_f \oplus \mathcal{J}_f$, so $w \notin \mathcal{F}_f \Rightarrow w \in \mathcal{J}_f$. 
+\end{proof}
+
+\subsubsection{Example of an interesting $w \in \mathcal{F}_f$}
+
+In this example, subsequences of the iterates are required, to illustrate the necessity of this argument in $\mathcal{J}_f$'s definition. \\
+\vspace{5mm}
+
+Consider $f(z) = z^2-1$. \\
+\vspace{2mm}
+Now take $w = -1$. \\
+\vspace{2mm}
+We will prove $w \in \mathcal{F}_f$ by choosing a neighbourhood of $w$ in which all sequences have a convergent subsequence on compact subsets of itself. \\
+\vspace{5mm}
+
+\begin{proof}
+
+We will consider the ball $\overline{B(w,\varepsilon)}$ for $0 < \varepsilon < \frac{1}{5}$. \\
+A maximal $\varepsilon$ satisfying this proof can be found \textcolor{blue}{\underline{\href{https://www.wolframalpha.com/input?i=positive+solution+of+\%CE\%B5+\%3D+\%282\%CE\%B5\%2B\%CE\%B5\%5E2\%29\%5E2}{here}}}. \\
+\vspace{5mm}
+
+$$\forall z_0 \in \overline{B(w,\varepsilon)}, \qquad z_0 = w+k_0e^{i \theta_0} \quad 0 \leq k_0 \leq \varepsilon, \quad \theta_0 \in [0, 2 \pi[$$
+and : \\
+\vspace{-3mm}
+\begin{align*}
+z_1 = f(z_0) &= z_0^2-1 \\
+&= (w+k_0 e^{i \theta_0})^2-1 \\
+&= 1 - 2k_0e^{i \theta_0} + k_0^2 e^{2i \theta_0}-1 \\
+&= -2k_0 \underbrace{\left( e^{i \theta_0}- \frac{k_0}{2} e^{2i \theta_0} \right)}_{\text{modulus } \leq 1+ \frac{k_0}{2} }
+\end{align*}
+\vspace{-2mm}
+It's then easy to see that $f(z_0) \in \overline{B(0,2 \varepsilon + \varepsilon^2)}$. \\
+\vspace{5mm}
+
+From $z_1 \in \overline{B(0, 2\varepsilon)}$, as $\varepsilon < \frac{1}{5}$ we have $\lvert z_1 \rvert < \frac{2}{5} + \frac{1}{25} = \frac{11}{25}$. \\
+Reusing the notation from earlier, we have $z_1 = 0 + k_1 e^{i \theta_1}$ for $0 \leq k_1 \leq 2\varepsilon + \varepsilon^2$ and $\theta_1 \in [0,2 \pi[$. \\
+$$f(z_1) = \underbrace{k_1^2}_{\leq \frac{121}{625} < \frac{125}{625} = \frac{1}{5}}e^{2i \theta_1}-1$$
+Which finally implies $f(z_1) \in \overline{B(w,\varepsilon)}$ again. \\
+\vspace{5mm}
+
+We have found a neighbourhood $B(w,\varepsilon)$ of $w$ in which any sequence of iterates has terms in one of or both of the sets $\overline{B(w, \varepsilon)}$ and $\overline{B(0, 2 \varepsilon)}$. \\
+\vspace{2mm}
+$\overline{B(w, \varepsilon)}$ is a the (compact) closure of the neighbourhood, so the Bolzano-Weierstrass theorem guarantees that we can take a subsequence of iterates (specifically such that we exclude all terms in $\overline{B(0, 2 \varepsilon)}$) which will converge (sequential compactness) in $\overline{B(w, \varepsilon)}$. \\
+\vspace{5mm}
+
+Thus we conclude $w \in \mathcal{F}_f$. 
+\end{proof}
+
+Note that this proof \textit{requires} the definition to extend to subsequences, because of how the given sequences of iterates of $f$ have values in disjoint sets. \\
+\vspace{2mm}
+
+Here is a visual representation of this disjointness (made interactive \textcolor{blue}{\underline{\href{https://www.geogebra.org/m/eee6ukbk}{here}}}) : \\
+\vspace{-3mm}
+\begin{center}
+\includegraphics[scale=0.9]{complementminus1} \\
+\textit{Fig. 1 - First terms of a sequence of sets $(b_n)$ where $b_n = f^n(B(w, \varepsilon))$}
+\end{center}
+
+As goes the proof, the maximal possible $\varepsilon$ for which the proof is valid is actually slightly greater than $\frac{1}{5}$ and can be obtained by solving $\varepsilon = (2 \varepsilon + \varepsilon^2)^2$ for $\varepsilon > 0$. Here is an image showing how this $\varepsilon$ appears in the Fatou component $w$ is in : \\
+\begin{center}
+\includegraphics[scale=0.3]{20231028105914}
+\textit{$Bas(0)$ (black) for $z \mapsto z^2-1$ with biggest neighbourhood of $w = -1$ that is still in $Bas(0)$ (red).} \\
+Image obtained using "Inverted lightness RGB" colourmap of my \textcolor{blue}{\underline{\href{https://github.com/ChrisMzz/pyogl-shader-tester/blob/main/shaders/frag_incriter.glsl}{Julia set shader}}}, with fixed $c$.
+\end{center}
+
+
+\bibliographystyle{alphaurl.bst}
+\bibliography{frctl_refs.bib}
+
+\end{document}

report/Annex/frctl_refs.bib

@@ -0,0 +1,49 @@
+
+@article{Jon19,
+ url = {https://theses.liacs.nl/1714},
+ author = {Jonckheere, L.S. (Luc) de},
+ publisher = {Thesis Bachelor Informatica LIACS},
+ title = {Efficiently Generating the Mandelbrot and Julia Sets},
+ year = {2019}
+},
+
+@misc{Sil13,
+ url = {https://solarianprogrammer.com/2013/02/28/mandelbrot-set-cpp-11/},
+ author = {Paul Silisteanu},
+ title = {The Mandelbrot set in C++11},
+ year = {2013}
+}
+
+
+@article{Sut14,
+author = {Sutherland, Scott},
+year = {2014},
+month = {01},
+pages = {37-60},
+title = {An Introduction to Julia and Fatou Sets},
+volume = {92},
+isbn = {978-3-319-08104-5},
+journal = {Springer Proceedings in Mathematics and Statistics},
+url={https://www.researchgate.net/publication/287394590_An_Introduction_to_Julia_and_Fatou_Sets}
+}
+
+@inproceedings{Lev97,
+  title={When do two rational functions have the same Julia set},
+  author={Genadi Levin and Feliks Przytycki},
+  year={1997},
+  url={https://api.semanticscholar.org/CorpusID:15478232}
+}
+
+@misc{Buz19,
+      title={Measures of maximal entropy for surface diffeomorphisms}, 
+      author={Jérôme Buzzi and Sylvain Crovisier and Omri Sarig},
+      year={2019},
+      eprint={1811.02240},
+      archivePrefix={arXiv},
+      primaryClass={math.DS}
+}
+
+
+
+
+

report/Annex/makefile

@@ -0,0 +1,18 @@
+# from https://tex.stackexchange.com/a/40759/297129
+
+
+.PHONY: frctl-art-annex.pdf all clean
+
+all: frctl-art-annex.pdf
+
+%.tex: %.raw
+	./raw2tex $< > $@
+
+%.tex: %.dat
+		./dat2tex $< > $@
+
+frctl-art-annex.pdf: frctl-art-annex.tex
+		latexmk -pdf -pdflatex="pdflatex -interaction=nonstopmode" -use-make frctl-art-annex.tex
+
+clean:
+	latexmk -CA