Clear[ik];
Clear[NK];
data = {x0 -> 0.7}
FX[Z_] := \[Sqrt]z
FX[z_] := -z^2 + 1
FX[z_] := Sin[-z]
FX[z_] := Abs[z^(2)] + 1
FX[z_] := Sin[z]
FX[z_] := -z^2 + 1
xx0 = x0 /. data
NK = 23
x[0] = xx0
PON[0] = ParametricPlot[{x[0], t}, {t, 0, FX[x[0]]}];
jk = 1;
While[jk < NK + 1, {x[jk] = FX[x[jk - 1]],
   PO[jk] = Plot[x[jk], {t, x[jk], x[jk - 1]}],
   PON[jk] = ParametricPlot[{x[jk], t}, {t, x[jk], FX[x[jk]]}]};
  jk++];
ik = 1
rxx = Table[{x[ik - 1], x[ik]}, {ik, NK}];
pl1 = Plot[x, {x, 0, 2}]
pl2 = ListPlot[rxx]
pl3 = Plot[FX[x], {x, 0, 2}]
POP = Table[PO[ik], {ik, 1, NK}];
POON = Table[PON[ik], {ik, 1, NK}];
s1 = Show[POP];
s2 = Show[POON];
Show[pl1, pl2, pl3, PlotRange -> {{0, 1}, {0, 1}}]
Show[pl1, s1, s2, pl3, PON[0], PlotRange -> {{0, 1}, {0, 1}}]
Show[pl1, s1, s2, pl3, PON[0], PlotRange -> All, Frame -> True]