Visualization of an iteration scheme in Mathematica

Question

I have a mapping T as attached (here. Now for $x_{0}\in C$ the Mann iteration process read as $x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}Tx_{n}$ n=0,1,2,3,... Now I want to obtain a visualization of the paper and plot given in the paper https://www.mdpi.com/2073-8994/11/12/1441

I know only the techniques that Mann iteration converges to a fixed point of a certain mapping as

a[n]=0.9;
x[0]=0.5;
T[x_]:=T[x]=x^2;
x[n]=(1-a[n-1])x[n-1]+x[n-1]T[x[n-1]]

But I do not to get figure like in the given paper. The Matlab code there is given but I do not like Matlab. I will be thankful if anyone can suggest me this implementation in mathematica.

Answer 1

I use the map in graphic. If I have misunderstood please let me know.

func[a_, b_] := Module[{f, rp, res},
  f[{x_, y_}] := 
   Piecewise[{{{2 - x, y}, 0 <= x < 1/7}, {{(x + 12)/7, y}, 
      1/7 <= x <= 2}}];
  rp = RandomPoint[Rectangle[{a, 0}, {b, 1/7}], 10000];
  res = NestList[f /@ # &, rp, 3];
  GraphicsRow[
   ListPlot[#, PlotRange -> {{0, 2}, {0, 1/7}}, 
      GridLines -> {{2 - 1/7, (1/7 + 12)/7}, None}, Frame -> True, 
      GridLinesStyle -> Red] & /@ res]]

Visualizing iterates for various point sets:

Column[{func[0, 1/7], func[1/7, 2], func[0, 2]}]