f[x_]:=a*x*(1-x);
g[x_]:=f[f[f[x]]];
r=(g[x]-x)/(f[x]-x)//Expand//FullSimplify

ContourPlot[r==0,{a,0,5},{x,0,1},PlotPoints->1000]

rx=D[r,x]//FullSimplify

NSolve[{r==0,rx==0},{a,x},Reals]

Solve[{r==0,rx==0},{a,x},Reals]

Solve[{r==0,rx==0},{a,x},Reals]//N