% !TeX TXS-program:compile = txs:///pdflatex

\documentclass[margin=5mm]{standalone}
\usepackage{ProfLycee}

\begin{document}

\begin{tikzpicture}[x=10cm,y=10cm,xmin=0,xmax=1.05,xgrille=1,xgrilles=0.5,ymin=0,ymax=0.65,ygrille=1,ygrilles=0.5]
	\AxesTikz[ElargirOx=0/0,ElargirOy=0/0]
	\AxexTikz{0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1}
	\AxeyTikz{0,0.1,0.2,0.3,0.4,0.5,0.6}
	%labels
	\draw (0.1,0) node[below=20pt,font=\small] {$u_0$} ;
	\draw[thick,densely dashed] (0.171,0.171)--(0.171,0) node[below=20pt,font=\small] {$u_1$} ;
	\draw[thick,densely dashed] (0.269,0.269)--(0.269,0) node[below=20pt,font=\small] {$u_2$} ;
	\draw[thick,densely dashed] (0.374,0.374)--(0.374,0) node[below=20pt,font=\small] {$u_3$} ;
	\draw[thick,densely dashed] (0.1,0.171)--(0,0.171) node[left=20pt,font=\small] {$u_1 = f\left(u_0\right)$} ;
	\ToileRecurrence[Fct={1.9*\x*(1-\x)},No=1,Uno=0.1,Nb=4,AffTermes=false]
	\draw[thick,magenta] (0.1,0)--(0.1,0.1);
	%courbe(s)
	\clip (\xmin,\ymin) rectangle (\xmax,\ymax) ;
	\draw[very thick,blue,samples=250,domain=0:1] plot (\x,{1.9*\x*(1-\x)}) ;
	\draw[blue] (0.8,0.3) node[above right,font=\large] {$\mathcal{C}_f$} ;
	\draw[CouleurVertForet,very thick] (0,0)--(0.65,0.65) node[below right,font=\large] {$D : y=x$} ;
\end{tikzpicture}

\end{document}