\(\(
f[x_] := 1 - a*x^2
		\)\)

a = 0.8

nmax = 30

y[0] = 0.5

Do[y[n + 1] = N[f[y[n]]], {n, 0, nmax}]

z[0] = y[0]

\(\(Do[z[4*l + 1] = y[l + 1]; z[4*l + 2] = y[l + 1]; 
      z[4*l + 3] = y[l + 1]; z[4*l + 4] = y[l + 1], {l, 0, nmax - 1}]
\)\)

t1 = Table[{z[2*m], z[2*m + 1]}, {m, 0, 2*(nmax - 1)}];

p1 = ListPlot[t1, PlotJoined -> True, PlotRange -> All]

t2 = Table[y[n], {n, 0, nmax}];

mi = Min[t2]

ma = Max[t2]

p2 = Plot[{x, f[x]}, {x, mi - 0.2, ma + 0.2}]

Show[p1, p2]