f[x_]:=a*x*(1-x)
g[x_]:=a*(1-2x)
x0=1.0/E;
ax=Flatten[Table[
x=Nest[f,x0,100];
Table[{a,x=f[x]},{100}]
,{a,0,4,0.001}],1]//Quiet;

ListPlot[ax]

n=1000;
aL=Table[
x=Nest[f,x0,100];
{a,1/n*Sum[Log[Abs[g[x=f[x]]]],{n}]}
,{a,0,4,0.001}]//Quiet;
ListPlot[aL]