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          Point[{0.5403418836722723`, 1.5418992601659907`, 0.5`}], 
          Point[{0.3499801664531856`, 1.5496205631671378`, 0.5`}], 
          Point[{0.3704917696485707`, 1.6641680590882422`, 0.5`}], 
          Point[{0.2675015457505166`, 1.1633108763890492`, 0.5`}], 
          Point[{0.6982739418335285`, 1.1266971395746588`, 0.5`}], 
          Point[{0.29914083509543193`, 1.5087218364910209`, 0.5`}], 
          Point[{0.47939840377314574`, 1.6351688753203328`, 0.5`}], 
          Point[{0.443927132414813`, 1.6096677579134604`, 0.5`}], 
          Point[{0.7707032626074283`, 0.8531753559129449`, 0.5`}], 
          Point[{0.6001785322431995`, 1.410523130567613`, 0.5`}], 
          Point[{0.5632715234557361`, 1.452649527319922`, 0.5`}], 
          Point[{0.6135474083587119`, 1.3376235026322367`, 0.5`}], 
          Point[{0.7648909037395023`, 0.8734680195632374`, 0.5`}], 
          Point[{0.5792877828872658`, 1.460150962089263`, 0.5`}], 
          Point[{0.4575632258147592`, 1.602330342351677`, 0.5`}], 
          Point[{0.7266008552424689`, 1.0038813134094593`, 0.5`}], 
          Point[{0.43672833095292135`, 1.6709940367948695`, 0.5`}], 
          Point[{0.6525088250212152`, 1.2332126977380524`, 
              0.5`}]};

resultat = {};
oldx = 0;
For[x = 1, x ≤  Dimensions[basepoint][[1]], 
      x = x + 1, 
tmpx = basepoint[[x]][[1]][[1]]; 
pointcourant = {oldx, tmpx}; 
oldx = tmpx; 
If[x > 1, 
        AppendTo[resultat, pointcourant]];
]

Dimensions[resultat][[1]]

resultat;

list1 = 
      ListPlot[resultat, AxesLabel -> {"x(n)", "x(n+1)"}, 
        PlotRange -> {{0.2, 0.8}, {0.2, 0.8}}, 
        ImageSize -> {400, 400} , 
        PlotStyle -> Hue[0.6]]

list2 = 
      ListPlot[{{0, 0}, {0.5, 0.5}, {0.8, 0.8}}, 
        AxesLabel -> {"x(n)", "x(n+1)"}, 
        PlotRange -> {{0.2, 0.8}, {0.2, 0.8}}, 
        ImageSize -> {400, 400} , PlotJoined -> True, 
        PlotStyle -> Hue[0]]

Show[list1, list2]