Finding a periodic orbit

Let’s suppose that I want to find a point of period 3 for the function f(x) = (x^3)-3x-1. Looking at graphs of y=x, y=f(x), and y=(f^3)(x), I notice that there might be such a point near x=-0.5.

f[x_] = x^3 - 3x - 1;
F[x_] = Nest[f,x,3];
Plot[{x,f[x],F[x]},{x,-1,1},
PlotRange -> {-2,1}]

I can use the FindRoot command to find it.

x0 = x /. FindRoot[F[x]==x,{x,-0.5}]

Let’s check:

NestList[f,x0,3]

Looks good!