I need to find fixed points of iterative map x[n] == 1/2 x[n-1]^2 - Mu.
My approach:
Subscript[g, n_ ][Mu_, x_] := Nest[0.5 * x^2 - Mu, x, n]
fixedPoints[n_] := Solve[Subscript[g, n][Mu, x] == x, x]
Plot[
Evaluate[{x,
Table[Subscript[g, 1][Mu, x], {Mu, 0.5, 4, 0.5}]}
], {x, 0, 0.5}, Frame -> True]
I'll change notation slightly (mostly so I myself can understand it). You might want something like this.
y[n_, mu_, x_] := Nest[#^2/2 - mu &, x, n]
fixedPoints[n_] := Solve[y[n, mu, x] == x, x]
The salient feature is that the "function" being nested now really is a function, in correct format.
Example:
fixedPoints[2]
Out[18]= {{x -> -1 - Sqrt[-3 + 2*mu]},
{x -> -1 + Sqrt[-3 + 2*mu]},
{x -> 1 - Sqrt[ 1 + 2*mu]},
{x -> 1 + Sqrt[ 1 + 2*mu]}}
Daniel Lichtblau
First of all, there is an error in your approach. Nest takes a pure function. Also I would use exact input, i.e. 1/2 instead of 0.5 since Solve is a symbolic rather than numeric solver.
Subscript[g, n_Integer][Mu_, x_] := Nest[Function[z, 1/2 z^2 - Mu], x, n]
Then
In[17]:= fixedPoints[1]
Out[17]= {{x -> 1 - Sqrt[1 + 2 Mu]}, {x -> 1 + Sqrt[1 + 2 Mu]}}
A side note:
Look what happens when you start very near to a fixed point (weird :) :
f[z_, Mu_, n_] := Abs[N@Nest[1/2 #^2 - Mu &, z, n] - z]
g[mu_] := f[1 + Sqrt[1 + 2*mu] - mu 10^-8, mu, 10^4]
Plot[g[mu], {mu, 0, 3}, PlotRange -> {0, 7}]
Edit
In fact, it seems you have an autosimilar structure there:
FrameLabelsin this case, as it tends to hide what you're asking for. Lastly, what exactly are you looking for here? Does your code work, or not? Is it to slow? What else have you tried? Please be specific.